Postscript - Courant Institute of Mathematical Sciences - NYU
Ergodic Theory of Infinite Dimensional Systems with Applications to Dissipative Parabolic PDEs Nader Masmoudi1
and
Lai-Sang Young2
Courant Institute of Mathematical Sciences 251 Mercer Street, New York, NY 10012 May 12, 2001
1
Introduction
This paper concerns the ergodic theory of a class of nonlinear dissipative PDEs of parabolic type. Leaving precise statements for later, we first give an indication of the nature of our results. We view the equation in question as a semi-group or dynamical system St on a suitable function space H, and assume the existence of a compact attracting set (as in Temam [15], Chapter 1). To this deterministic system, we add a random force in the form of a “kick” at periodic time intervals, defining a Markov chain X with state space H. We assume that the combined effect of the semi-group and our kicks sends balls to compact sets. Under these conditions, the existence of invariant measures for X is straightforward. The goal of this paper is a better understanding of the set of invariant measures and their ergodic properties. In a state space as large as ours, particularly when the noise is bounded and degenerate, the set of invariant measures can, in principle, be very large. In this paper, we discuss two different types of conditions that reduce the complexity of the situation. The first uses the fact that for the type of equations in question, high modes tend to be contracted. By actively driving as many of the low modes as needed, we show that the dynamics resemble those of Markov chains on RN with smooth transition probabilities. In particular, the set of ergodic invariant measures is finite, and every aperiodic ergodic measure is exponentially mixing. The second type of conditions we consider is when all of the Lyapunov exponents of X are negative. As in finite dimensions, we show under these conditions that nearby orbits cluster together in a phenomenon known as “random sinks”. The conditions in the last paragraph give a general understanding of the structure of invariant measures; they alone do not guarantee uniqueness. (Indeed, it is not the case that for the equations in question, invariant measures are always unique; 1 2
Email [email protected] Email [email protected] . This research is partially supported by a grant from the NSF
1
see Theorem 3.) For uniqueness, one needs to guarantee that there are places for distinct ergodic components to meet. To this end, we have identified some conditions expressed in terms of existence of special sequences of controls. These conditions are quite special; however, they are easily verified for the equations of interest. Assuming these conditions, the uniqueness of the invariant measure follows readily. In the case of negative Lyapunov exponents, there is, in fact, a stronger form of uniqueness or stability, namely that all solutions independent of initial conditions become asymptotically close to one other as time goes to infinity. This work is inspired by a number of recent papers on the uniqueness of invariant measure for the Navier-Stokes equations ([5], [1], [3], [7], [8]), and by [4], which proves uniqueness of invariant measure for a different equation. With the exception of [7] and [8], all of the other authors worked with unbounded noise. Naturally there is overlap among these papers and with the first part of ours. More detailed references will be given as the theorems are stated. Instead of working directly with specific PDEs, we have elected to prove our ergodic theory results for general randomly perturbed dynamical systems on infinite dimensional Hilbert spaces satisfying conditions compatible with the PDEs of interest. This allows us to make more transparent the relations between the various dynamical properties and the mechanisms responsible for them. Once our “abstract” results are in place, to apply them to specific equations, it suffices to verify that the conditions in the theorems are met. (In this regard, we are influenced by [7], which takes a similar approach.) This paper is organized as follows. Before proceeding to a discussion of our “abstract results”, we first give a sample of their applications. This is done in Section 2. Sections 3 and 4 treat the two types of conditions that lead to simpler structures for invariant measures. In each case, we begin with a general discussion and finish with proofs of concrete results for PDEs which we now state.
2
Statement of Results for PDEs
This section contains precise formulations of results on PDEs that can be deduced from our “abstract theory”. The theorems below are proved in Sects. 3.3 and 4.3.
2.1
The Navier-Stokes system
The first application of our general results is to the 2-D incompressible Navier-Stokes equations in the 2−torus T2 = (R/2πZ)2 . We consider the randomly forced system P∞ ∂t u − ν∆u + u · ∇u = −∇p + k=1 δ(t − k)ηk (x) , (1) div(u) = 0, u(t = 0) = u0 2
R where u0 (x) ∈ L2 (T2 ), div(u0 ) = 0, u0 = 0, ν > 0 is the viscosity, and where the ηk ’s are i.i.d. random fields which can be expanded as ηk (x) =
∞ X
bj ξjk ej (x) .
(2)
j=1
Here the Hilbert space in question is 2
2
H = {u, u ∈ L (T ), div(u) = 0, and
Z
u = 0} ,
and {ej , j ≥ 1} is the orthonormal basis consisting of the eigenfunctions of the Stokes operator −∆ej + ∇pj = λj ej , div(ej ) = 0, with λ1 ≤ λ2 ≤ · · · . We assume that ξjk , j, k ∈ N, are independent random variables where ξjk is distributed according to a law which has a positive Lipschitz density ρj with to the Lebesgue measure P respect 2 2 on [−1, 1]. Finally, the bj are required to satisfy ∞ b = a < ∞ for some a > 0. j=1 j From (1), we define a Markov chain uk with values in H given by uk = u(k + 0, ·). That is to say, if St is the semi-group generated by the unforced Navier-Stokes equaP tion, i.e. equation (1) without the term ∞ δ(t − k)ηk (x), and S = S1 , then k=1 uk+1 = S(uk ) + ηk .
Theorem 1 (Uniqueness of invariant measure and exponential mixing). For the system above, there exists N ≥ 1 depending only on the viscosity ν and on a such that if bj 6= 0 for all 1 ≤ j ≤ N, then the Markov chain uk has a unique invariant measure µ in H. Moreover, for all u0 ∈ H, the distribution Θk of uk converges exponentially fast to µ in the sense that for every test function f : H → R of class C 0,σ , σ > 0, there exists C = C(f, u0 ) such that for all k ≥ 1, Z Z f dΘk − f dµ < Cτ k for some τ < 1 depending only on the H¨older exponent σ.
Papers [7] and [8] together contain a proof of the uniqueness of invariant measure part of Theorem 1; these papers rely on ideas different from ours. While this manuscript was being written, we received electronic preprints [9] and [10] which together prove the results in Theorem 1 using methods similar to ours. Theorem 1’. The result 1 holds if we replace L2 by H s , any s ∈ N, and P∞in Theorem impose the restriction j=1 λsj b2j = a2 < ∞ on the noise.
Remark. In the theorems above, we can also treat noises that are bounded but not compact provided that we consider the Markov chain uk = u(k − 0, ·) or, equivalently, 3
uk+1 = S(uk + ηk ). An example of bounded, noncompact noise satisfying the conditions of Theorems 1 and 1’ is the following: Let VN be the span of {e1 , e2 , ..., eN }, and consider N X ηk = bj ξjk ej + ηk′ (3) j=1
where bj 6= 0 for all j, 1 ≤ j ≤ N, and ηk′ are i.i.d. random variables with a law supported on a bounded set in VN⊥ . Our next result gives a stronger form of uniqueness than the previous one. It guarantees, under the assumption of negative Lyapunov exponents, that independent of initial condition, all the solutions eventually come together and evolve as one, their time evolution depending only on the realization of the noise. Lyapunov exponents are defined in Sect. 4.1. Having only negative Lyapunov exponents means, roughly speaking, that infinitesimally the semi-group is contractive on average along typical orbits. More regularity is required for the next result; thus we work in H 2 . Let A(0) ⊂ H 2 denote the closure of the set of points accessible under the Markov chain uk with u0 = 0. Theorem 2 (Asymptotic uniqueness of solutions independent of initial conditions). Consider the system defined by (1) with H = H 2 and where the ηk are i.i.d. with a law which has bounded support. Assume there is an invariant measure µ supported on A(0) such that all of its Lyapunov exponents are strictly negative. Then (a) µ is the unique invariant measure the Markov chain uk has in H; (b) there exists λ < 0 such that for almost every sequence of ηk and every pair of initial conditions u0 , u′0 ∈ H, there exists C = C(u0 , u′0) such that if uk+1 = S(uk ) + ηk and u′k+1 = S(u′k ) + ηk for all k ≥ 0, then kuk − u′k k ≤ Ceλk
∀k≥0.
Observe that for this result very little is required of the structure of the noise. Remark. We will explain in Sect. 4.4 (see Remark) that for fixed positive viscosity, S is a uniform contraction near 0, and so it continues to be a contraction for sufficiently small bounded noise. Very small but unbounded noise is treated in [12]. As noise level increases, it is likely that there is a range where S is no longer a contraction but all of its Lyapunov exponents remain negative. Indeed, for any ergodic invariant measure µ of the Navier-Stokes system, the largest Lyapunov exponent λ1 is either < 0, = 0, or > 0: λ1 > 0 can be interpreted as “temporal chaos”; λ < 0 implies the asymptotic uniqueness of solutions as we have shown; the case λ1 = 0 is sometimes regarded as less significant because it can often be perturbed away. Of these three possibilities, the only one that has been proved to occur is λ1 < 0. 4
Remark. Theorems 1 and 2 apply to other nonlinear parabolic equations for which all solutions of the unforced equation relax to their unique stable stationary solutions.
2.2
The real Ginzburg-Landau equation
Our second application is to the following equation, which, following [4], we refer to as the real Ginzburg-Landau equation. We consider a periodic domain in one space dimension, i.e. T = (R/2πZ), and consider the system P∞ ∂t u − ν∆u − u + u3 = k=1 δ(t − k)ηk (x) , (4) u(t = 0) = u0 .
Here H = L2 (T), {ej , j ≥ 0} is the orthonormal basis defined by −∆ej = λj ej , λ1 ≤ λ2 ≤ · · · , ν > 0 is a positive constant, and the ηk ’s are i.i.d. random fields which can be expanded as ∞ X bj ξjk ej (x) . (5) ηk (x) = j=0
We assume the same conditions on ξjk and bj as in the first paragraph of Sect. 2.1. The unforced equation in (4) is somewhat more unstable than the (unforced) Navier-Stokes equation. It has at least three stationary solutions: two stable ones, namely u = 1 and u = −1, and an unstable one, namely u = 0. Our next result shows that the number of invariant measures vary depending on how localized the forcing is, particularly in the zeroth mode. Theorem 3 (Number of ergodic measures). Consider the Markov chain uk defined by the system in (4). P 2 2 2 (a) There exists α > 0 such that if ∞ j=0 |bj | = a ≤ α , then there are at least two different invariant measures. (b) There exists N depending only on ν and on a such that if bj 6= 0 for all 0 ≤ j ≤ N, then the number of ergodic invariant measures is finite. (c) If bj 6= 0 for all 0 ≤ j ≤ N and b0 > 1, then the invariant measure is unique, and for every initial condition u0 ∈ H, the distribution of uk converges to it exponentially fast in the sense of Theorem 1. In contrast to part (a), we observe that to obtain uniqueness of the invariant measure, we may take bj , 1 ≤ j ≤ N, to be arbitrarily small as long as they are > 0, and the forcing in the zeroth mode, i.e. b0 ξ0k , can be arbitrarily weak as long as its law has a tail which extends beyond [−1, 1]. As will be explained in Sect. 3.4, the condition b0 > 1 above can, in fact, be replaced by b0 > κ for a smaller κ. Theorem 3 complements [4], which drives high rather than low modes, and proves uniqueness for unbounded noise using techniques very different from ours. 5
3 3.1
Invariant Measures and their Ergodic Properties Formulation of abstract results
Setting and notation. Let S : H → H be a transformation of a separable Hilbert space H, and let ν be a probability measure on H. We consider the Markov chain X = {un , n = 0, 1, 2, · · · } on H defined by either (I) un+1 = S(un ) + ηn
or
(II) un+1 = S(un + ηn )
where η0 , η1 , · · · are i .i .d . with law ν. The following notation is used throughout this paper: BH (R) or simply B(R) denotes the ball of radius R in H, i.e. B(R) = {u ∈ H, kuk ≤ R}; K denotes the support of ν; and given an initial distribution Θ0 of u0 , the distribution of un under X is denoted by Θn . If T : H → H is a mapping and µ is a measure on H, then T∗ µ is the measure defined by (T∗ µ)(E) = µ(T −1(E)). Standing Hypotheses (P1) (a) S(B(R)) is compact ∀R > 0; (b) ∀R > 0, ∃MR > 0 such that ∀u, v ∈ B(R), kSu − Svk ≤ MR ku − vk. (P2) ∀a > 0, ∃R0 = R0 (a) such that if K ⊂ B(a), then ∀R > 0, ∃N0 = N0 (R) ∈ Z+ such that for u0 ∈ B(R), un ∈ B(R0 ) ∀n ≥ N0 . (P3) ∃γ < 1 such that given R > 0, there is a finite dimensional subspace V ⊂ H such that if PV and PV ⊥ denote orthogonal projections from H onto V and V ⊥ respectively, then ∀u, v ∈ B(R), kPV ⊥ S(u) − PV ⊥ S(v)k ≤ γku − vk. (P4) (a) K is compact if X is defined by (I), bounded if X is defined by (II). (b) Let V be given by (P3) with R = R0 . Then ν = (PV )∗ ν × (PV ⊥ )∗ ν where (PV )∗ ν has a density ρ with respect to the Lebesgue measure on V , Ω := {ρ > 0} has piecewise smooth boundary and ρ|Ω is Lipschitz. We remark that (P1)–(P3) are selected to reflect the properties of general (nonlinear) parabolic PDEs. Definition 3.1 A probability measure µ on H is called an invariant measure for X if Θ0 = µ implies Θn = µ for all n > 0. Lemma 3.1 Assume (P1), (P2) and (P4)(a). Then (i) X has an invariant measure; (ii) there exists a compact set A ⊂ B(R0 ) on which all invariant measures of X are supported.
6
Proof. Let A0 = B(R0 ). For n > 0, let An = S(An−1 ) + K in the case of (I) and An = S(An−1 +K) in the case of (II). Then each An is compact, and by (P2), An ⊂ A0 for all n ≥ some N0 . Let 0 −1 A = ∪N (∩∞ k=0 AkN0 +i ) . i=0
Then A is compact, contained in B(R0 ), and satisfies S(A) + K = A. To construct an invariant measure for X , pick an arbitrary u0 ∈ A, and let Θ0P= δu0 , the Dirac measure at u0 . Then any accumulation point of the sequence { n1 i 0, there exists τ = τ (σ) < 1 such that the following holds for every f : H → R of class C 0,σ : for µ-a.e. u0 , there exists C = C(f, u0 ) such that Z Z < Cτ n for all n ≥ 1. f dΘn − f dµ Let X n denote the n-step Markov chain associated with X .
Theorem A (Structure of invariant measures). Assume (P1)–(P4). Then (1) X has at most a finite number of ergodic invariant measures. (2) If (X n , µ) is ergodic for all n ≥ 1, then (X , µ) is exponentially mixing for H¨older continuous observables. The reasons behind these results are that under (P1)–(P4), X resembles a Markov chain on RN whose transition probabilities have densities. One expects, therefore, the same type of decomposition into ergodic and mixing components. We now give a condition that guarantees the uniqueness of invariant measures and other convergence properties. This condition is expressed in terms of the existence of special sequences of controls; it is quite strong, but is easily verified for the PDEs under consideration.
7
(C) Given ε0 > 0 and R > 0, there is a finite sequence of controls ηˆ0 , · · · ηˆn such that for all u0 , u′0 ∈ B(R), if uk+1 = S(uk ) + ηˆk and u′k+1 = S(u′k ) + ηˆk for k < n, then kun − u′n k < ε0 . Theorem B (Sufficient condition for uniqueness and mixing). (P1)–(P4) and (C). Then
Assume
(1) X has a unique invariant measure µ, and (X , µ) is exponentially mixing; (2) ∃τ = τ (σ) < 1 such that ∀f ∈ C 0,σ and for every u0 ∈ H, there exists C s.t. Z Z f dΘn − f dµ < Cτ n for all n ≥ 1.
Recalling that the invariant measure µ is supported on a (relatively small) compact subset of H, we remark that the assertion in (2) above is considerably stronger than the usual notion of exponential mixing: it tells us about initial conditions far away from the support of µ. This property is reminiscent of the idea of Sinai-RuelleBowen measures for attractors in finite dimensional dynamical systems.
3.2
Proofs of abstract results (Theorems A and B)
We will prove Theorems A and B for the case where X is defined by (I); the proofs for (II) are very similar. Also, to avoid the obstruction of main ideas by technical details, we will assume (PV )∗ ν is the normalized Lebesgue measure on Ω := {u ∈ V, kuk ≤ r} for some r > 0; the general case is messier but conceptually not different. Let M = MR0 where R0 is given by (P2) and MR0 is as defined in (P1). The following notation is used heavily: Given u0 and η = (η0 , η1 , η2 , · · · ) ∈ K N , we define ui(η) inductively by letting u0 (η) = u0 and ui (η) = S(ui−1 (η)) + ηi−1 for i > 0. Notation such as ui (η0 , · · · , ηn−1 ) for a finite sequence (η0 , · · · , ηn−1 ) with i ≤ n has the obvious meaning, as does u′i(η) for given u′0 . Lemma 3.2 (Matching Lemma) Let δ = r(2M)−1 . There is a set Γ ⊂ K N with ν N (Γ) > 0 such that ∀u0 , u′0 ∈ B(R0 ) with ku0 −u′0 k < δ, there is a measure-preserving map Φ : Γ → K N with the property that ∀η ∈ Γ, kun (η) − u′n (Φ(η))k ≤ ku0 − u′0 kγ n
∀n ≥ 0 .
By virtue of (P4)(b), K = Ω × E where Ω ⊂ V is as above and E ⊂ V ⊥ . We write η0 = (η01 , η02 ) with η01 ∈ Ω, η02 ∈ E. Since all of our operations take place in V , it is convenient to introduce the notation Kε := {u ∈ Ω, kuk ≤ ε} × E, so that in particular Kr = K.
8
Proof. Suppose ku0 − u′0 k < δ. We define Φ(1) : K r2 = Kr−M δ → H by 1
2
(η0′ , η0′ ) = Φ(1) (η0 ) := (η01 + PV S(u0 ) − PV S(u′0 ), η02 ) . Observe that (i) kη0′ 1 k < (r − Mδ) + M · ku0 − u′0 k < r, so that Φ(1) (K 2r ) ⊂ K; (ii) Φ(1) preserves ν-measure; and (iii) for η0 ∈ K 2r , if u1 = u1 (η0 ) and u′1 = u′1 (Φ(1) (η0 )), then PV u1 = PV u′1
and
kPV ⊥ u1 − PV ⊥ u′1 k < γku0 − u′0 k.
We may, therefore, repeat the argument above with (u1 , u′1) in the place of (u0 , u′0 ), defining for each u1 = u1 (η0 ), η0 ∈ K 2r , a map from Kr−M δγ = Kr(1− 1 γ) to K. Put 2 together, this defines an injective map Φ(2) : K 2r × Kr(1− 1 γ) → K 2 which carries ν 2 2 measure to ν 2 -measure such that for each (η0 , η1 ) ∈ K r2 × Kr(1− 1 γ) , if u2 = u2 (η0 , η1 ) 2 and u′2 = u′2 (Φ(2) (η0 , η1 )), then PV u2 = PV u′2 and kPV ⊥ u2 − PV ⊥ u′2 k < γ 2 ku0 − u′0 k. Continued ad infinitum, this process defines a map Φ : Γ := K 2r × Kr(1− 1 γ) × Kr(1− 1 γ 2 ) × · · · → K N 2
2
with the desired properties. Clearly, ν(Γ) = Πi≥0 (1 − 21 γ i )D > 0 where D =dimV . Proof of Theorem A(1). Recall that ifPµ is an ergodic invariant measure for X , then by the Birkhoff Ergodic Theorem, n1 0n−1 δui (η) → µ for µ-a.e. u0 and ν N -a.e. η = (η0 , η1 , · · · ). This together with Lemma 3.2 implies that if µ, and µ′ are ergodic measures and there exist u0 ∈ supp(µ) and u′0 ∈ supp(µ′ ) with ||u0 − u′0 k < δ, then µ = µ′ . Since all invariant measures of X are supported on the compact set A (Lemma 3.1), it follows that there cannot be more than a finite number of them. Proof of the uniqueness of invariant measure part of Theorem B. From the last paragraph, we know that all the ergodic components of µ are pairwise ≥ δ apart in distance. Thus condition (C) with ε0 = δ and R = R0 guarantees that there is at most one ergodic component. We remark that the uniqueness of invariant measure results in Theorem 1 and Theorem 3(c) follow immediately from the preceding discussion once the abstract hypotheses (P1)–(P4) and (C) are checked for these equations. The next lemma is used only to prove the general result in Theorem A(2); it is not needed for the applications in Theorems 1–3. (Both the Navier-Stokes and Ginzberg-Landau equations satisfy much stronger conditions, making this argument unnecessary.) Let B(u, ε) denote the ball of radius ε centered at u, and let P n (·|u) denote the n-step transition probability given u. In the language introduced earlier, if Θ0 = δu , then P n (·|u) = Θn (·). 9
Lemma 3.3 Let µ be an invariant measure with the property that (X n , µ) is ergodic for all n ≥ 1. We fix B = B(˜ u, ε˜) where u˜ ∈ supp µ and ε˜ > 0. Then there exist + N0 ∈ Z and α0 > 0 such that P N0 (B|u) ≥ α0 for every u ∈ supp µ. Proof. Pick arbitrary u0 ∈ suppµ. Until nearly the end of the proof, the discussion ˆ n defined by pertains to this one point. Consider the “restricted distribution” Θ ˆ n (G) = ν n {(η0 , · · · , ηn−1 ) : ηi ∈ Kr−M δγ i ∀i < n and un (η0 , · · · , ηn−1 ) ∈ G} Θ ˆ n. where δ is as in Lemma 3.2, and let Wn denote the support of Θ Claim 1. d(u0, ∪n>0 Wn ) = 0. Pn−1 ˆ ˆ i converges weakly Proof. By compactness, a subsequence of n1 i=0 (Θi (A))−1 Θ to a probability measure µ ˜ on A (where A is as in Lemma 3.1). Since the restrictions on ηi become milder and milder as i → ∞, µ ˜ is an invariant measure for X . By ˆ construction, all the Θi are supported on supp µ, so we must have µ ˜ = µ, for we know from Theorem A(1) that all the other ergodic invariant measures have their supports bounded away from supp µ. Let N = N(u0 ) be such that d(u0 , WN ) < ε where ε < δ is a small positive number to be determined. Claim 2. For all k ≥ 0 and u ∈ WkN , ∃u′ ∈ W(k+1)N such that ku − u′ k < γ kN ε. Proof. The claim is true for k = 0 by choice of N. We prove it for k = 1: Let ∈ WN be such that ku0 −u′0 k < ε, and fix an arbitrary u ∈ WN . By definition, there exist ηi ∈ Kr−M δγ i such that u = uN (η0 , · · · , ηN −1 ). We wish to use the proximity ′ of u′0 to u0 and the Matching Lemma to produce (η0′ , · · · , ηN −1 ) with the property ′ ′ ′ ′ N that uN (η0 , · · · , ηN −1 ) ∈ W2N and kuN − uN k < εγ . To obtain the first property, it is necessary to have ηi′ ∈ Kr−M δγ i+N for all i < N. We proceed as follows: since ku0 − u′0 k < ε and η0 ∈ Kr−M δ , ∃η0′ ∈ Kr−M δ+M ε such that ku1 (η0 ) − u′1 (η0′ )k < εγ; similarly ∃η1′ ∈ Kr−M δγ+M εγ such that ku2 (η0 , η1 ) − u′1 (η0′ , η1′ )k < εγ 2 , and so on. (See the proof of Lemma 3.2.) Thus ηi′ ∈ Kr−M γ i (δ−ε) , and assuming ε is sufficiently small that δγ N < (δ − ε), we have ηi′ ∈ Kr−M δγ i+N . To prove the assertion for k = 2, we pick an arbitrary u ∈ W2N , which, by definition, is equal to vN from some v0 ∈ WN . Since we have shown that there exists v0′ ∈ W2N with kv0 − v0′ k < γ N ε, it suffices to ′ ′ repeat the argument above to obtain vN ∈ W3N with kvN − vN k < γ 2N ε. ˆ kN (B(˜ Claim 3. There exists k1 = k1 (u0) s.t. for k ≥ k1 , P kN (B|u) ≥ Θ u, 2ε˜ )) > 0 for all u ∈ H with ku − u0 k < δ. u′0
Proof. Let N (W, ε) denote the P ε-neighborhood of W ⊂ H. If follows from Claim 2 that if NkN := N (WkN , 2ε ki=0 γ iN ), then NkN ⊂ N(k+1)N for all k. Moreover, the ergodicity of (X N , µ) together with an observation similar to that in Claim 1 shows that the closure of ∪k NkN contains suppµ. Thus NkN ∩ B(˜ u, 4ε˜ ) 6= ∅ for large P∞ iN ˆ kN (B(˜ enough k. If 2ε i=1 γ < 4ε˜ , then Θ u, 2ε˜ )) > 0. Now for u with ku − u0 k < δ, 10
ˆ n starting from u0 can be coupled to a part of the the entire restricted distribution Θ (unrestricted) distribution starting from u. Thus for sufficiently large n, P n (B|u) ≥ ˆ n (B(˜ Θ u, 2ε˜ )). (1)
(n)
To finish, we cover supp µ with a finite number of δ-balls centered at u0 , · · · , u0 , ˆ where kˆ1 = maxi k1 (u(i) ) and N ˆ = Πi N(u(i) ). The lemma is and choose N0 = kˆ1 N 0 0 ˆ N0 (B(˜ ˆ N0 is the restricted distribution starting proved with α0 = mini Θ u, 2ε˜ )) where Θ (i) from u0 . From Lemma 3.2, we see that associated with each pair of points (u0 , u′0) with ku0 − u′0 k < δ, there is a cascade of matchings between un and u′n , leading to the definition of a measure-preserving map Φ : Γ := K r2 × Kr(1− 1 γ) × Kr(1− 1 γ 2 ) × · · · → K N 2
2
with the property that for η ∈ Γ, kui (η) − u′i (Φ(η))k ≤ γ i ku0 − u′0 k
for all i ≤ n.
The main goal in the next proof is, in a sense, to extend Φ to all of K N by attempting repeatedly to match the orbits that have not yet been matched. Proof of Theorem A(2). We consider for simplicity the case N0 = 1. Let u0 , u′0 ∈ supp µ, and let Θn and Θ′n denote the distributions of un and u′n respectively. We seek to define a measure-preserving map Φ : K N → K N and to estimate the difference between Θn and Θ′n by Z Z Z ′ |f (un (η)) − f (u′n (Φ(η)))| dν N (η) . In := f dΘn − f dΘn ≤
Let B be a ball of diameter δ centered at some point in suppµ. By Lemma 3.3, P (B|u0) ≥ α0 , and P (B|u′0 ) ≥ α0 . Matching u1 ∈ B to u′1 ∈ B, we define a measure˜ 1 → K for some Γ ˜ 1 ⊂ K with |Γ ˜ 1 | = α0 . This extends, by the preserving map Φ(1) : Γ ˜ 1 × Γ → K N . The map Matching Lemma, to a measure-preserving map Φ : Γ1 = Γ Φ|Γ1 represents the cascade of future couplings initiated by Φ(1) . Suppose now that Φ has been defined on ∪k≤n Γk where Γk is the set of η matched at step k. More precisely, Γ1 , Γ2 , · · · , Γn are disjoint subsets of K N , and each Γk is of ˜ k × Γ for some Γ ˜ k ⊂ K k ; the matching of uk and u′ in B that takes the form Γk = Γ k ˜ k → K k , while the cascade of future matchings place at step k defines a map Φ(k) : Γ ˜ k × Γ → K N . We now explain how to initiated by Φ(k) results in the definition of Φ : Γ (n) (n) ˜ n = K n \ ∪k≤n Γ where Γ = Γ ˜ k × Γ(n−k−1) is the first n-factors define Γn+1 . Let G k k ˜ n+1 defined by (η0 , · · · , ηn−1 ) ∈ G ˜ n; in Γk . Consider the restricted distribution Θ ′ ˜ the corresponding distribution Θn+1 is defined similarly. By Lemma 3.3, an α0 fraction of these two distributions can be matched, defining an immediate matching 11
˜ n+1 → K n+1 with Γ ˜ n+1 ⊂ G ˜ n × K and |Γ ˜ n+1 | = α0 |G ˜ n |. Future couplings Φ(n+1) : Γ (n+1) N ˜ n+1 × Γ. that result from Φ define Φ : Γn+1 → K with Γn+1 = Γ n ˜ We claim that ν (Gn ) decreases exponentially. This requires a little argument, for even though at each step a fraction of α0 of what is left is matched, our matchings (n) are “leaky”, meaning not every orbit defined by a sequence in Γk can be matched to ˜ n ), we write K N \∪k≤n Γk something reasonable at the (n+1)st step. To estimate ν n (G N ˜ n × K . The dynamics of (Gn , Hn ) → as the disjoint union Gn ∪ Hn where Gn = G (Gn+1 , Hn+1) are as follows: An α0 -fraction of Gn leaves Gn at the next step; of this part, a fraction of Πi≥0 (1 − 12 γ i )D (recall that D is the dimension of V ) goes into Γn+1 (see Lemma 3.2) while the rest goes into Hn+1 . At the same time, a fraction of Hn returns to Gn+1 . We claim that this fraction is bounded away from zero for all n. (n) To see this, consider one Γk at a time, and observe (from the definition of Γk ) that (n) (n+1) ˜ k |γ n−k . |(Γk × K) \ Γk | ∼ const |Γ Combinatorial Lemma Let a0 , b0 > 0, and suppose that an and bn satisfy recursively an+1 ≥ (1 − α0 )an + α1 bn
and
bn+1 ≤ (1 − α1 )bn + α0 an
for some 0 < α0 , α1 < 1. Then there exits c > 0 such that
an bn
> c for all n.
The proof of this purely combinatorial lemma is left as an exercise. We deduce ˜ n ) ≤ Cβ n for some C > 0 and from it that inf n |Gn |/|Hn | > 0, which implies ν n (G β < 1. This in turn implies that |Γn+1| ≤ Cβ n . Proceeding to the final count, we let f : supp µ → R be such that |f | < C1 and |f (u) − f (v)| < C1 ku − vkσ . Then In ≤
Z
˜n G
n
Z
|f (un (η0 , · · · , ηn−1 ))|dν + |f (u′n (η0 , · · · , ηn−1 ))|dν n (n) (n) n K −Φ (∪k≤n Γk ) X Z |f (u(η0 , · · · , ηn−1 )) − f (u′n (Φ(n) (η0 , · · · , ηn−1)))|dν n (6) + (n)
k≤n
Γk
≤ 2C1 · Cβ n +
X
Cβ k−1 · C1 (δγ n−k )σ
k≤n
≤ const n · [max(β, γ σ )]n ≤ const · τ n . Since these estimates are uniform for all pairs u0 , u′0, we obtain by integrating over u′0 that Z Z ≤ const · τ n . f dΘn − f dµ
12
Proof of Theorem B. We will prove, in the next paragraph, that assertion (2) in Theorem B holds for any invariant measure µ of X . From this (1) follows immediately: since (X , µ) is exponentially mixing, it is ergodic; and since µ is chosen arbitrarily, it must be the unique invariant measure. To prove the claim above, we pick arbitrary u0 ∈ H, u′0 ∈ A, and compare their distributions Θn and Θ′n as we did in the proof of Theorem A(2). First, by waiting a suitable period, we may assume that Θn is supported in B(R0 ) (where R0 is as in (P2)). By condition (C) with ε0 = δ where δ is as in Lemma 3.2, there is a set of controls of length N0 and having ν N0 -measure α0 for some α0 R> 0 that steer the R ˆ n| entire ball B(R0 ) into a set of diameter < δ. The estimate for | f dΘn − f dΘ now proceeds as in Theorem A(2), with the use of these special controls taking the place of Lemma 3.3 to guarantee that an α0 -fraction of what is left is matched every N0 steps. Averaging u′0 with respect to µ, we obtain the desired result.
3.3
Applications to PDEs: Proofs of Theorems 1 and 3
In this subsection, we prove the theorems related to PDEs stated in Sect. 2.1. Proof of Theorem 1. We will prove that the abstract hypotheses (P1)–(P4) and (C) hold for the incompressible Navier-Stokes equation in L2 for the type of noise specified. Let S(u0) = u(t = 1) where u is the solution of the Navier-Stokes equation with initial data u0 , and let uk = S(uk−1 ) + ηk . Most of the computations below are classically known (see for instance [2], [14]); we include them for completeness. We start by recalling a few properties of the Navier-Stokes equation in the 2-D torus. First, the following energy estimate holds for all t > 0: Z t 1 1 2 (7) ||u(t)||L2 + ν ||∇u||2L2 = ||u0||2L2 . 2 2 0 R Since u = 0, we have the Poincare inequality ||∇u||L2 ≥ ||u||L2 .
(8)
||S(u)||L2 ≤ e−ν ||u||L2 ;
(9)
From (7) and (8), it follows that
thus (P2) is satisfied by taking R0 (a) > 1−e1−ν a. On the other hand, for any two solutions u and v with initial conditions u0 and v0 , we have Z 1 2 2 ∂t ||u − v||L2 + ν||∇(u − v)||L2 ≤ | (u − v).∇v(u − v)| 2 (10) ≤ C||∇v||L2 ||u − v||L2 ||u − v||H 1 C ν ||u − v||2H 1 + ||∇v||2L2 ||u − v||2L2 . ≤ 2 ν 13
(H¨older and Sobolev inequalities are used to get the second line, and the CauchySchwartz inequality is used to get the third.) Then, applying a Gronwall lemma, we get Z 1
||S(u0) − S(v0 )||2L2 + ν
0
||(u − v)(s)||2H 1 ds ≤ CR ||u0 − v0 ||2L2 .
(11)
Here and below, CR denotes a generic constant depending only on R, an upper bound on the L2 norm of u0 , and on the viscosity ν. (P1)(b) follows from (11). To prove that (P3) holds, we use (11), (7) and a Chebychev inequality to deduce the existence of a time s, 0 < s < 1, such that ν||(u − v)(s)||2H 1 ≤ 4CR ||u0 − v0 ||2L2 , ν||u(s)||2H 1 < 2R2 and ν||v(s)||2H 1 < 2R2 . Combining these estimates with energy estimates in H 1 for t > s, namely, Z t 1 1 2 (12) ||∇u(t)||L2 + ν ||∆u||2L2 = ||∇u(s)||2L2 , 2 2 s Z t 1 1 2 (13) ||∇v(t)||L2 + ν ||∆v||2L2 = ||∇v(s)||2L2 , 2 2 s 1 (14) ∂t ||u − v||2H 1 + ν||u − v||2H 2 ≤ ||u − v||H 2 ||u − v||H 1 ||u||H 2 + ||v||H 2 2 1 ν ||u − v||2H 2 + ||u||2H 2 + ||v||2H 2 ||u − v||2H 1 , ≤ 4 ν integrating (14) between s and 1 and using again a Gronwall lemma, we deduce easily that (15) ||S(u0) − S(v0 )||H 1 ≤ CR ||u0 − v0 ||L2 . For any γ > 0 and R > 0, we may take N large enough that if VN :=span{e1 , e2 , ..., eN }, then CR ||u||L2 ≤ γ||u||H 1 ∀u ∈ VN⊥ . This together with (15) proves (P3). Finally, property (C) is satisfied by taking ηi = 0 for 1 ≤ i ≤ n0 where n0 is large enough that Re−νn0 ≤ ε0 (see (9)). The product structure of the noise ν 3 in property (P4)(b) holds because ξjk in (2) are independent; the assumption on PV ∗ ν holds because bj 6= 0 for 1 ≤ j ≤ N where N is as in (P3) and the law for ξjk has density ρj . Proof of Theorem 1’. We now prove (P1)–(P4) and (C) in H s . To prove (P1)(b), we use the energy estimates 1 ∂t ||u||2H s + ν||u||2H s+1 ≤ C||u||H s ||u||H s+1 ||u||H 1 2 C ν ||u||2H s+1 + ||u||2H 1 ||u||2H s , ≤ 2 ν 3
(16)
We hope our dual use of the symbol ν as viscosity and as noise does not lead to confusion.
14
1 ∂t ||u − v||2H s + ν||u − v||2H s+1 ≤ C||u − v||H s ||u − v||H s+1 (||u||H s+1 + ||v||H s+1 ) (17) 2 ν C ≤ ||u − v||2H s+1 + (||u||2H s+1 + ||v||2H s+1 )||u − v||2H s , 2 ν and Gronwall’s lemma between times 0 and 1. To prove (P3), we proceed as in the case of L2 , showing the existence of a time τ , 0 < τ < 1, such that ||(u−v)(τ )||H s+1 ≤ 4CR ||u0 −v0 ||H s and ||u(τ )||H s+1 , ||v(τ )||H s+1 ≤ 4CR where ||u0 ||H s , ||v0||H s < R. Then using (16) and (17) with s replaced by s + 1 and integrating between τ and 1, we deduce that ||S(u0) − S(v0 )||H s+1 ≤ CR ||u0 − v0 ||H s ,
(18)
from which we obtain (P3). To prove (P2), we make use of the regularizing effect of the Navier-Stokes equation in 2-D (19) ||S(u0)||H s ≤ Cs (||u||L2 ) where Cs is a function depending only on s (see [14]). Since BH s (a) ⊂ BL2 (a), we know from (P3) for L2 that if u0 ∈ BL2 (R), we have un ∈ BL2 (R0 ) ∀n ≥ some N0 . Taking Rs = Cs (R0 ) + a, we get that un ∈ BH s (Rs ) ∀n ≥ N0 . To prove (C), we argue as in L2 , taking ηi = 0, 1 ≤ i ≤ n0 , for large enough n0 and appealing to the fact that Cs (r) → 0 as r → 0. We remark that (P2) and (C) above can be proved directly without going through L2 . Next we move on to the real Ginzburg-Landau equation. Proof of Theorem 3. For simplicity, we take ν = 1. (a) We need to prove that there exist two disjoint stable sets A1 and A−1 , stable in the sense that ∀u ∈ A±1 , S(u) + η ∈ A±1 ∀η ∈ K. Let A1 = {u ∈ H, ||u − 1||L2 ≤ β}
(20)
where β is a constant to be determined. We recall for each φ ∈ R the energy estimate Z 1 2 2 (21) ∂t ||u − φ||L2 + ||∇(u − φ)||L2 + u(u − 1)(u + 1)(u − φ) dx = 0 . 2 T Substituting φ = 1 in (21), we get 1 ∂t ||u − 1||2L2 + ||∇(u − 1)||2L2 ≤ − 2
Z
u(u + 1)(u − 1)2 dx.
(22)
T
Now for any φ with 0 < φ < 1, we have u(u + 1)(u − 1)2 ≥ φ(φ + 1)(u − 1)2 u(u + 1)(u − 1)2 ≥ −1 15
if
u ≥ φ or u ≤ −1 − φ, ∀u.
(23)
Hence Z Z 2 u(u + 1)(u − 1) dx ≥ (1{u≥φ} + 1{u≤−1−φ} )φ(φ + 1)(u − 1)2 − meas{u ≤ φ}. T
T
Since the first term on the right side is Z 2 1{−1−φ 1 can be relaxed.
4 4.1
Dynamics with Negative Lyapunov Exponents Formulation of abstract results
We consider a semi-group St on H and a Markov chain X defined by (I) or (II) in the beginning of Sect. 3.1. In order for Lyapunov exponents to make sense, we need to impose differentiability assumptions. (P1’) (a) S(B(R)) is compact ∀R > 0; (b) S is C 1+Lip , meaning for every u ∈ H, there exists a bounded linear operator Lu : H → H with the property for all h ∈ H, 1 lim {S(u + εh) − S(u) − Lu (εh)} = 0 ε→0 ε
(34)
and ∀R > 0, ∃MR such that ∀u, v ∈ B(R), kLu − Lv k ≤ MR ku − vk. Since Lemma 3.1 clearly holds with (P1) replaced by (P1’), we let A be as in Section 3. 17
Proposition 4.1 Assume (P1’), (P2) and (P4)(a), and let µ be an invariant measure for X . Then there is a measurable function λ1 on H with −∞ ≤ λ1 < ∞ such that for µ-a.e. u0 and ν N -a.e. η = (η0 , η1 , η2 , · · · ), lim
n→∞
1 log kLun−1 ◦ · · · ◦ Lu1 ◦ Lu0 k = λ1 (u0 ) . n
Moreover, λ1 is constant µ-a.e. if (X , µ) is ergodic. This proposition follows from a direct application of the Subadditive Ergodic Theorem [6] together with the boundedness of kLu k on A (see also Lemma 4.1 below). We will refer to the function or, in the ergodic case, number λ1 as the top Lyapunov exponent of (X , µ). This section is concerned with the dynamics of X when λ1 < 0. We begin by stating a result, namely Theorem C, which gives a general description of the dynamics when λ1 < 0. This result, however, is not needed for our application to PDEs. The proof of Theorem 2 uses only Theorem D, which is independent of Theorem C. Let µ be an invariant measure of X . Theorem C concerns the conditional measures of µ given the past. That is to say, we view X as starting from time −∞, i.e. consider · · · , u−2 , u−1, u0 , u1 , u2, · · · defined by un+1 = Sun + ηn ∀n ∈ Z where · · · , η−2 , η−1 , η0 , η1 , η2 , · · · are ν-i.i.d. Then for ν Z -a.e. η = (· · · , η−1 , η0 , η1 , · · · ), the conditional probability of µ given η − := (· · · , η−2 , η−1 ) is well defined. We denote it by µη . Theorem C (Random sinks). Assume (P1’), (P2) and (P4)(a), and let µ be an ergodic invariant measure with λ1 < 0. Then there exists k0 ∈ Z+ such that for ν Z -a.e. η ∈ K Z , µη is supported on exactly k0 points of equal mass. This result is well known for stochastic flows in finite dimensions (see [11]). In the next theorem we impose a condition slightly stronger than (C) in Sect. 3.1 to obtain the type of uniqueness result needed for Theorem 2. (C’) There exists uˆ0 ∈ H such that for all ε0 > 0 and R > 0, there is a finite sequence of controls ηˆ0 , · · · ηˆn such that for all u0 ∈ B(R), if uk+1 = Suk + ηˆk and uˆk+1 = S uˆk + ηˆk for all k < n, then kun − uˆn k < ε0 . For u ∈ H, we define the accessibility set A(u) as follows: let A0 (u) = {u}, An (u) = S(An−1 (u)) + K for n > 0, and A(u) = ∪n≥0 An (u). Theorem D (Asymptotic uniqueness of solutions independent of initial condition). Assume (P1’), (P2), (P4)(a) and (C’). Suppose there is an ergodic invariant measure µ supported on A(ˆ u0 ) for which λ1 < 0. Then µ is the only invariant measure X has, and the following holds for ν N -a.e. η = (η0 , η1 , · · · ): ∀u0 , u′0 ∈ H,
kun (η) − u′n (η)k ≤ Ceλn 18
∀n > 0
where λ is any number > λ1 and C = C(u0 , u′0, λ). Roughly speaking, Theorem D allows us to conclude that all the orbits are eventually “the same” once we know that the linearized flows along some orbits are contractive. This passage from a local to a global phenomenon is made possible by condition (C’), which in the abstract is quite special but is satisfied by a number of standard parabolic PDEs.
4.2
Proofs of abstract results (Theorems C and D)
Let A be the compact set in Lemma 3.1, and let K denote the support of ν as before. We consider the dynamical system F : K N × A → K N × A defined by F (η, u) = (ση, S(u) + η0 ) where η = (η0 , η1 , η2 , · · · ) and σ is the shift operator, i.e. σ(η0 , η1 , η2 , · · · ) = (η1 , η2 , · · · ). The following is straightforward. Lemma 4.1 Let µ be an invariant measure of X in the sense of Definition 3.1. Then F preserves ν N × µ, and (F, ν N × µ) is ergodic if and only if (X , µ) is ergodic in the sense of Definition 3.2. Our next lemma relates the top Lyapunov exponent of a system, which describes the average infinitesimal behavior along its typical orbits, to the local behavior in neighborhoods of these orbits. A version applicable to our setting is contained in [13]. Let B(u, α) = {v ∈ H, kv − uk < α}. Proposition 4.2 [13] Let µ be an invariant measure, and assume that λ1 < 0 µ-a.e. Then given ε > 0, there exist measurable functions α, γ : K N × A → (0, ∞) and a measurable set Λ ⊂ K N × A with (ν N × µ)(Λ) = 1 such that for all (η, u0 ) ∈ Λ and v0 ∈ B(u0 , α(η, u0)), kvn (η) − un (η)k < γ(η, u0) e(λ1 +ε)n
∀n ≥ 0 .
We first prove Theorem D, from which Theorem 2 is derived. Proof of Theorem D. From (P2), it follows that we need only to consider initial conditions in B(R0 ). Fix ε > 0 and let α and Λ be as in Proposition 4.2 for the dynamical system (F, ν N × µ). We make the following choices: 99 . Covering (1) Let α0 > 0 be a number small enough that (ν N × µ){α > 2α0 } > 100 1 the compact set A(ˆ u0 ) with a finite number of 2 α0 -balls, we see that there exists u˜0 ∈ A(ˆ u0 ) such that
Γ1 := {η ∈ K N : B(˜ u0 , α0 ) ⊂ B(u, α(η, u)) for some u with (η, u) ∈ Λ} has positive ν N -measure. 19
(2) Since u˜0 ∈ A(ˆ u0 ), there is a sequence of controls (˜ η0 , · · · , η˜k−1) that puts uˆ0 in B(˜ u0 , 21 α0 ). Choose δ > 0 and Γ2 ⊂ K k with ν k (Γ2 ) > 0 such that if u0 ∈ B(ˆ u0 , δ) and (η0 , · · · , ηk−1) ∈ Γ2 , then uk (η0 , · · · , ηk−1 ) ∈ B(˜ u0 , α0 ). (3) Condition (C’) guarantees that there exists a sequence of controls (ˆ η0 , · · · , ηˆj−1 ) 1 j that puts the entire ball B(R0 ) inside B(ˆ u0 , 2 δ). Choose Γ3 ⊂ K with ν j (Γ3 ) > 0 such that every sequence (η0 , · · · , ηj−1) ∈ Γ3 puts B(R0 ) inside B(ˆ u0 , δ). Let Γ ⊂ K N be the set defined by {(η0 , · · · , ηj−1) ∈ Γ3 ; (ηj , · · · , ηj+k−1) ∈ Γ2 ; (ηj+k , ηj+k+1, · · · ) ∈ Γ1 } . Clearly, ν N (Γ) > 0. The following holds for ν N -a.e. η : Fix η, and let Bn denote the nth image of B(R0 ) for this sequence of kicks. By the ergodicity of (σ, ν N ), there exists N such that σ N η ∈ Γ. Choosing N ≥ N0 (R0 ), we have, by (P2), that BN ⊂ B(R0 ). The choice in (3) then guarantees that BN +j ⊂ B(ˆ u0 , δ), and the choice in (2) guarantees that BN +j+k ⊂ B(˜ u0 , α0 ). By (1), BN +j+k ⊂ B(u, α(u, σ N +j+k η)) N +j+k for some u with (σ η, u) ∈ Λ. Proposition 4.2 then says that when subjected to the sequence of kicks defined by σ N +j+k η, all orbits with initial conditions in BN +j+k converge exponentially to each other as n → ∞. Hence this property holds for all orbits starting from B(R0 ) when subjected to η. Theorem D is proved. Proceeding to Theorem C, the measures µη defined in Sect. 4.1 are called the sample or empirical measures of µ. They have the interpretation of describing what one sees at time 0 given that the system has experienced the sequence of kicks η − = (· · · , η−2 , η−1 ). The characterization of µη in the next lemma is useful. We introduce the following notation: Let Sη0 : H → H be the map defined by Sη0 (u) = Su + η0 ; for a measure µ on H, Sη0 ∗ µ is the measure defined by (Sη0 ∗ µ)(E) = µ(Sη−1 E). 0 Lemma 4.2 Let µ be an invariant measure for X . Then for ν Z -a.e. (· · · , η−2 , η−1 , η0 , ...), (Sη−1 Sη−2 · · · Sη−n )∗ µ converges weakly to µη . Proof: Fix a continuous function ϕ : A → R, and define ϕ(n) : K Z → R by Z (n) ϕ (η) = ϕ d((Sη−1 Sη−2 · · · Sη−n )∗ µ) Z = ϕ(Sη−1 Sη−2 · · · Sη−n (u))dµ(u).
η =
(35)
−n −n Then ϕ(n) is B−1 -measurableR where B−1 is the σ-algebra on K Z generated by coor−n+1 dinates η−1 , · · · , η−n . Since Sη−n ∗ µ dν(η−n ) = µ, we have E(ϕ(n) |B−1 ) = ϕ(n−1) . (n) The martingale convergence theorem then tells us that ϕ convergence ν Z -a.e. to −∞ a function measurable on B−1 . It suffices to carry out the argument above for a countable dense set of continuous functions ϕ.
20
Lemma 4.3 Given δ > 0, ∃N = N(δ) ∈ Z+ such that for ν Z -a.e. η, there is a set Eη consisting of ≤ N points such that µη (Eη ) > (1 − δ). Proof: Let α and γ be the functions in Proposition 4.2 for the dynamical system (F, ν N × µ). Given δ > 0, we let α0 , γ0 > 0 be constants with the property that if G = {(η, u) : α(η, u) ≥ α0 , γ(η, u) ≤ γ0 } and Γ = {η ∈ K N : µ{u : (η, u) ∈ G} > 1 − δ} , then ν N (Γ) > 1 − δ. Consider η ∈ K Z such that (i) µη = lim(Sη−1 Sη−2 · · · Sη−n )∗ µ and (ii) (η−n , η−n+1, · · · ) ∈ Γ for infinitely many n > 0. By Lemma 4.2 and the ergodicity of (σ, ν Z ), we deduce that the set of η satisfying (i) and (ii) has full measure. We will show that the property in the statement of the lemma holds for these η. Fix a cover {B1 , · · · , BN } of A by α20 -balls, and let η be as above. We consider n arbitrarily large with (η−n , η−n+1 , · · · ) ∈ Γ. For each i, 1 ≤ i ≤ N, such that Bi ∩ {u ∈ H : ((η−n , η−n+1 , · · · ), u) ∈ G} = 6 ∅, pick an arbitrary point u(i) in this set. Our choices of G and Γ ensure that µ(∪i B(u(i) , α0 )) > 1 − δ, and that the diameter of (Sη−1 Sη−2 · · · Sη−n )B(u(i) , α0 ) is ≤ γ0 α0 e(λ+ε)n . We have thus shown that a set of µη -measure > 1 − δ is contained in ≤ N balls each with diameter ≤ γ0 α0 e(λ+ε)n . The result follows by letting n → ∞. To prove Theorem C, we need to work with a version of (F, ν N ×µ) that has a past. Let F˜ : K Z × A → K Z × A be such that F˜ : (η, u) 7→ (ση, Sη0 u), and let ν Z ∗ µ be the measure which projects onto ν Z in the first factor and has conditional probabilities µη on η-fibers. That ν Z ∗ µ is F˜ -invariant follows immediately from Lemma 4.2. It is also easy to see that (F˜ , ν Z ∗ µ) is ergodic if and only if (F, ν N × µ) is. Proof of Theorem C. It follows from Lemma 4.3 that for ν Z -a.e. η, µη is atomic, with possibly a countable number of atoms. We now argue that there exists k0 ∈ Z+ such that for a.e. η, µη has exactly k0 atoms of equal mass. Let h(η) = supu∈H µη {u} . To see that h is a measurable function on K Z , let P (n) , n = 1, 2, · · · , be an increasing sequence of finite measurable partitions of A such that diamP (n) → 0 as n → ∞. Then for each P ∈ P (n) , η 7→ µη (P ) is a measurable function, as are hn := maxP ∈P (n) µη (P ) and h := limn hn . Observe that h(ση) ≥ h(η), with > being possible in principle since Sη0 is not necessarily one-to-one. However, the measurability of h together with the ergodicity of (σ, ν Z ) implies that h is constant a.e. Let us call this value h0 . From the last lemma we know that h0 > 0. 21
To finish, we let X = {(η, u) ∈ K Z × A : µη {u} = h0 }. Then X is a measurable set, (ν Z ∗ µ)(X) > 0 and F˜ −1 X ⊃ X. This together with the ergodicity of (F˜ , ν Z ∗ µ) implies that (ν Z ∗ µ)(X) = 1, which is what we want.
4.3
Application to PDEs: Proof of Theorem 2
Let St be the semi-group generated by the (unforced) Navier-Stokes system, and let S = S1 . Lemma 4.4 S is C 1+Lip in H 2 (R2 ). Proof. It is easy to see that Lu is defined by Lu w = ψ(1) where ψ is the solution of the linear problem ∂t ψ + U.∇ψ + ψ.∇U − ν∆ψ = −∇p, (36) ψ(t = 0) = w , divψ = 0 where U denotes the solution of the Navier-Stokes system with initial data u. That Lu is linear, continuous and goes from H 2 to H 2 is obvious. To prove that (34) holds, let U and V be the solutions of the Navier-Stokes system with initial data u and u + ǫw respectively. Then y = V − U − ǫψ satisfies ∂t y + (U + ǫψ).∇y + y.∇V + ǫ2 ψ.∇ψ − ∆y = −∇p (37) y(t = 0) = 0 , div(y) = 0. By a simple computation, we get that ||y(t = 1)||H 2 ≤ C(1 + ||w||2H 2 )ǫ2 , where here and below C denotes a constant depending only on the H 2 norm of u. To prove that Lu is Lipschitz, i.e., ||(Lu − Lv )w||H 2 ≤ C||u − v||H 2 ||w||H 2 ,
(38)
we define Lv w = φ(1) where φ solves an equation analogous to (36) with V in the place of U, V being the solution with initial condition v. The desired estimate ||(ψ − φ)(t = 1)||H 2 is obtained by subtracting this equation from (36). Remark. We observe here that the top Lyapunov exponent is negative if the noise is sufficiently small. We will show, in fact, that given any positive viscosity ν, if a (see Sect. 2.1 for definition) is small enough, then S : H 2 → H 2 is a contraction on ν the ball of radius 2C . Rewriting equations (16) and (17) with s = 2, we have ∂t ||u||2H 2 + ν||u||2H 3 ≤
22
C2 ||u||4H s , ν
(39)
∂t ||u − v||2H 2 + ν||u − v||2H 3 ≤
C2 (||u||2H 3 + ||v||2H 3 )||u − v||2H 2 . ν
(40)
ν ν For u0 with ||u0||H 2 ≤ 2C and a noise with a ≤ 2C (1 − e−ν/4 ), it follows from (39) and a Gronwall lemma that ν 2 ||S(u0)||2H 2 ≤ e−ν/2 , 2C
from which we obtain ||u1 ||H 2 ≤ ν
ν . 2C
Z
0
Moreover, from (39), we have that
1
||u||2H 3 ≤
ν3 , 16C 2
so that if v is another solution of the Navier-Stokes system with ||v0 ||H 2 ≤ (40) gives ν ||u − v||2H 2 ≤ ||u0 − v0 ||2H 2 e−ν+ 8 .
ν , 2C
then
Proof of Theorem 3. It suffices to check the hypotheses of Theorem D: (P1’) is proved in Lemma 4.4, and we explained in the proof of Theorem 1’ why (C’) holds with uˆ0 = 0.
References [1] J. Bricmont, A. Kupiainen and R. Lefevere, Exponential mixing for the 2D Navier-Stokes dynamics, 2000 preprint. [2] P. Constantin and C. Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. x+190 pp. [3] W. E, J. Mattingly and Ya. G. Sinai, Gibbsian Dynamics and ergodicity for the stochastically forced 2D Navier-Stokes equation, 2000 preprint. [4] J.-P. Eckmann and M. Hairer, Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise, 2000 preprint. [5] F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equations, NoDEA 1 (1994), 403-426. [6] J. F. C. Kingman, Subadditive processes, Ecole d’´et´e des Probabilit´es de Saint-Flour, Lecture Notes in Math., 539, Springer (1976) [7] S. Kuksin and A. Shirikyan, Stochastic dissipative PDEs and Gibbs measures, Commun. Math. Phys. 213 (2000), no. 2, 291–330. 23
[8] S. Kuksin and A. Shirikyan, On dissipative systems perturbed by bounded random kick-forces, 2000 preprint. [9] S. Kuksin and A. Shirikyan, A coupling approach to randomly forced nonlinear PDE’s, 2001 preprint. [10] S. Kuksin, A. L. Piatnitski and A. Shirikyan, A coupling approach to randomly forced nonlinear PDEs. II, 2001 preprint [11] Y. Le Jan, Equilibre statistique pour les produits de diffeomorphismes aleatoires independants, Ann. Inst. Henri Poincare (Probabilites et Statistiques), 23, (1987), 111-120. [12] J. Mattingly, Ergodicity of 2D Navier-Stokes Equations with random forcing and large viscosity, Commun. Math. Phys., 206 (1999), 273-288. [13] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. Math., 115, (1982), 243-290. [14] R. Temam, Navier-Stokes equations and nonlinear functional analysis. Second edition. CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. xiv+141 pp. [15] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. xxii+648 pp.
24
and
Lai-Sang Young2
Courant Institute of Mathematical Sciences 251 Mercer Street, New York, NY 10012 May 12, 2001
1
Introduction
This paper concerns the ergodic theory of a class of nonlinear dissipative PDEs of parabolic type. Leaving precise statements for later, we first give an indication of the nature of our results. We view the equation in question as a semi-group or dynamical system St on a suitable function space H, and assume the existence of a compact attracting set (as in Temam [15], Chapter 1). To this deterministic system, we add a random force in the form of a “kick” at periodic time intervals, defining a Markov chain X with state space H. We assume that the combined effect of the semi-group and our kicks sends balls to compact sets. Under these conditions, the existence of invariant measures for X is straightforward. The goal of this paper is a better understanding of the set of invariant measures and their ergodic properties. In a state space as large as ours, particularly when the noise is bounded and degenerate, the set of invariant measures can, in principle, be very large. In this paper, we discuss two different types of conditions that reduce the complexity of the situation. The first uses the fact that for the type of equations in question, high modes tend to be contracted. By actively driving as many of the low modes as needed, we show that the dynamics resemble those of Markov chains on RN with smooth transition probabilities. In particular, the set of ergodic invariant measures is finite, and every aperiodic ergodic measure is exponentially mixing. The second type of conditions we consider is when all of the Lyapunov exponents of X are negative. As in finite dimensions, we show under these conditions that nearby orbits cluster together in a phenomenon known as “random sinks”. The conditions in the last paragraph give a general understanding of the structure of invariant measures; they alone do not guarantee uniqueness. (Indeed, it is not the case that for the equations in question, invariant measures are always unique; 1 2
Email [email protected] Email [email protected] . This research is partially supported by a grant from the NSF
1
see Theorem 3.) For uniqueness, one needs to guarantee that there are places for distinct ergodic components to meet. To this end, we have identified some conditions expressed in terms of existence of special sequences of controls. These conditions are quite special; however, they are easily verified for the equations of interest. Assuming these conditions, the uniqueness of the invariant measure follows readily. In the case of negative Lyapunov exponents, there is, in fact, a stronger form of uniqueness or stability, namely that all solutions independent of initial conditions become asymptotically close to one other as time goes to infinity. This work is inspired by a number of recent papers on the uniqueness of invariant measure for the Navier-Stokes equations ([5], [1], [3], [7], [8]), and by [4], which proves uniqueness of invariant measure for a different equation. With the exception of [7] and [8], all of the other authors worked with unbounded noise. Naturally there is overlap among these papers and with the first part of ours. More detailed references will be given as the theorems are stated. Instead of working directly with specific PDEs, we have elected to prove our ergodic theory results for general randomly perturbed dynamical systems on infinite dimensional Hilbert spaces satisfying conditions compatible with the PDEs of interest. This allows us to make more transparent the relations between the various dynamical properties and the mechanisms responsible for them. Once our “abstract” results are in place, to apply them to specific equations, it suffices to verify that the conditions in the theorems are met. (In this regard, we are influenced by [7], which takes a similar approach.) This paper is organized as follows. Before proceeding to a discussion of our “abstract results”, we first give a sample of their applications. This is done in Section 2. Sections 3 and 4 treat the two types of conditions that lead to simpler structures for invariant measures. In each case, we begin with a general discussion and finish with proofs of concrete results for PDEs which we now state.
2
Statement of Results for PDEs
This section contains precise formulations of results on PDEs that can be deduced from our “abstract theory”. The theorems below are proved in Sects. 3.3 and 4.3.
2.1
The Navier-Stokes system
The first application of our general results is to the 2-D incompressible Navier-Stokes equations in the 2−torus T2 = (R/2πZ)2 . We consider the randomly forced system P∞ ∂t u − ν∆u + u · ∇u = −∇p + k=1 δ(t − k)ηk (x) , (1) div(u) = 0, u(t = 0) = u0 2
R where u0 (x) ∈ L2 (T2 ), div(u0 ) = 0, u0 = 0, ν > 0 is the viscosity, and where the ηk ’s are i.i.d. random fields which can be expanded as ηk (x) =
∞ X
bj ξjk ej (x) .
(2)
j=1
Here the Hilbert space in question is 2
2
H = {u, u ∈ L (T ), div(u) = 0, and
Z
u = 0} ,
and {ej , j ≥ 1} is the orthonormal basis consisting of the eigenfunctions of the Stokes operator −∆ej + ∇pj = λj ej , div(ej ) = 0, with λ1 ≤ λ2 ≤ · · · . We assume that ξjk , j, k ∈ N, are independent random variables where ξjk is distributed according to a law which has a positive Lipschitz density ρj with to the Lebesgue measure P respect 2 2 on [−1, 1]. Finally, the bj are required to satisfy ∞ b = a < ∞ for some a > 0. j=1 j From (1), we define a Markov chain uk with values in H given by uk = u(k + 0, ·). That is to say, if St is the semi-group generated by the unforced Navier-Stokes equaP tion, i.e. equation (1) without the term ∞ δ(t − k)ηk (x), and S = S1 , then k=1 uk+1 = S(uk ) + ηk .
Theorem 1 (Uniqueness of invariant measure and exponential mixing). For the system above, there exists N ≥ 1 depending only on the viscosity ν and on a such that if bj 6= 0 for all 1 ≤ j ≤ N, then the Markov chain uk has a unique invariant measure µ in H. Moreover, for all u0 ∈ H, the distribution Θk of uk converges exponentially fast to µ in the sense that for every test function f : H → R of class C 0,σ , σ > 0, there exists C = C(f, u0 ) such that for all k ≥ 1, Z Z f dΘk − f dµ < Cτ k for some τ < 1 depending only on the H¨older exponent σ.
Papers [7] and [8] together contain a proof of the uniqueness of invariant measure part of Theorem 1; these papers rely on ideas different from ours. While this manuscript was being written, we received electronic preprints [9] and [10] which together prove the results in Theorem 1 using methods similar to ours. Theorem 1’. The result 1 holds if we replace L2 by H s , any s ∈ N, and P∞in Theorem impose the restriction j=1 λsj b2j = a2 < ∞ on the noise.
Remark. In the theorems above, we can also treat noises that are bounded but not compact provided that we consider the Markov chain uk = u(k − 0, ·) or, equivalently, 3
uk+1 = S(uk + ηk ). An example of bounded, noncompact noise satisfying the conditions of Theorems 1 and 1’ is the following: Let VN be the span of {e1 , e2 , ..., eN }, and consider N X ηk = bj ξjk ej + ηk′ (3) j=1
where bj 6= 0 for all j, 1 ≤ j ≤ N, and ηk′ are i.i.d. random variables with a law supported on a bounded set in VN⊥ . Our next result gives a stronger form of uniqueness than the previous one. It guarantees, under the assumption of negative Lyapunov exponents, that independent of initial condition, all the solutions eventually come together and evolve as one, their time evolution depending only on the realization of the noise. Lyapunov exponents are defined in Sect. 4.1. Having only negative Lyapunov exponents means, roughly speaking, that infinitesimally the semi-group is contractive on average along typical orbits. More regularity is required for the next result; thus we work in H 2 . Let A(0) ⊂ H 2 denote the closure of the set of points accessible under the Markov chain uk with u0 = 0. Theorem 2 (Asymptotic uniqueness of solutions independent of initial conditions). Consider the system defined by (1) with H = H 2 and where the ηk are i.i.d. with a law which has bounded support. Assume there is an invariant measure µ supported on A(0) such that all of its Lyapunov exponents are strictly negative. Then (a) µ is the unique invariant measure the Markov chain uk has in H; (b) there exists λ < 0 such that for almost every sequence of ηk and every pair of initial conditions u0 , u′0 ∈ H, there exists C = C(u0 , u′0) such that if uk+1 = S(uk ) + ηk and u′k+1 = S(u′k ) + ηk for all k ≥ 0, then kuk − u′k k ≤ Ceλk
∀k≥0.
Observe that for this result very little is required of the structure of the noise. Remark. We will explain in Sect. 4.4 (see Remark) that for fixed positive viscosity, S is a uniform contraction near 0, and so it continues to be a contraction for sufficiently small bounded noise. Very small but unbounded noise is treated in [12]. As noise level increases, it is likely that there is a range where S is no longer a contraction but all of its Lyapunov exponents remain negative. Indeed, for any ergodic invariant measure µ of the Navier-Stokes system, the largest Lyapunov exponent λ1 is either < 0, = 0, or > 0: λ1 > 0 can be interpreted as “temporal chaos”; λ < 0 implies the asymptotic uniqueness of solutions as we have shown; the case λ1 = 0 is sometimes regarded as less significant because it can often be perturbed away. Of these three possibilities, the only one that has been proved to occur is λ1 < 0. 4
Remark. Theorems 1 and 2 apply to other nonlinear parabolic equations for which all solutions of the unforced equation relax to their unique stable stationary solutions.
2.2
The real Ginzburg-Landau equation
Our second application is to the following equation, which, following [4], we refer to as the real Ginzburg-Landau equation. We consider a periodic domain in one space dimension, i.e. T = (R/2πZ), and consider the system P∞ ∂t u − ν∆u − u + u3 = k=1 δ(t − k)ηk (x) , (4) u(t = 0) = u0 .
Here H = L2 (T), {ej , j ≥ 0} is the orthonormal basis defined by −∆ej = λj ej , λ1 ≤ λ2 ≤ · · · , ν > 0 is a positive constant, and the ηk ’s are i.i.d. random fields which can be expanded as ∞ X bj ξjk ej (x) . (5) ηk (x) = j=0
We assume the same conditions on ξjk and bj as in the first paragraph of Sect. 2.1. The unforced equation in (4) is somewhat more unstable than the (unforced) Navier-Stokes equation. It has at least three stationary solutions: two stable ones, namely u = 1 and u = −1, and an unstable one, namely u = 0. Our next result shows that the number of invariant measures vary depending on how localized the forcing is, particularly in the zeroth mode. Theorem 3 (Number of ergodic measures). Consider the Markov chain uk defined by the system in (4). P 2 2 2 (a) There exists α > 0 such that if ∞ j=0 |bj | = a ≤ α , then there are at least two different invariant measures. (b) There exists N depending only on ν and on a such that if bj 6= 0 for all 0 ≤ j ≤ N, then the number of ergodic invariant measures is finite. (c) If bj 6= 0 for all 0 ≤ j ≤ N and b0 > 1, then the invariant measure is unique, and for every initial condition u0 ∈ H, the distribution of uk converges to it exponentially fast in the sense of Theorem 1. In contrast to part (a), we observe that to obtain uniqueness of the invariant measure, we may take bj , 1 ≤ j ≤ N, to be arbitrarily small as long as they are > 0, and the forcing in the zeroth mode, i.e. b0 ξ0k , can be arbitrarily weak as long as its law has a tail which extends beyond [−1, 1]. As will be explained in Sect. 3.4, the condition b0 > 1 above can, in fact, be replaced by b0 > κ for a smaller κ. Theorem 3 complements [4], which drives high rather than low modes, and proves uniqueness for unbounded noise using techniques very different from ours. 5
3 3.1
Invariant Measures and their Ergodic Properties Formulation of abstract results
Setting and notation. Let S : H → H be a transformation of a separable Hilbert space H, and let ν be a probability measure on H. We consider the Markov chain X = {un , n = 0, 1, 2, · · · } on H defined by either (I) un+1 = S(un ) + ηn
or
(II) un+1 = S(un + ηn )
where η0 , η1 , · · · are i .i .d . with law ν. The following notation is used throughout this paper: BH (R) or simply B(R) denotes the ball of radius R in H, i.e. B(R) = {u ∈ H, kuk ≤ R}; K denotes the support of ν; and given an initial distribution Θ0 of u0 , the distribution of un under X is denoted by Θn . If T : H → H is a mapping and µ is a measure on H, then T∗ µ is the measure defined by (T∗ µ)(E) = µ(T −1(E)). Standing Hypotheses (P1) (a) S(B(R)) is compact ∀R > 0; (b) ∀R > 0, ∃MR > 0 such that ∀u, v ∈ B(R), kSu − Svk ≤ MR ku − vk. (P2) ∀a > 0, ∃R0 = R0 (a) such that if K ⊂ B(a), then ∀R > 0, ∃N0 = N0 (R) ∈ Z+ such that for u0 ∈ B(R), un ∈ B(R0 ) ∀n ≥ N0 . (P3) ∃γ < 1 such that given R > 0, there is a finite dimensional subspace V ⊂ H such that if PV and PV ⊥ denote orthogonal projections from H onto V and V ⊥ respectively, then ∀u, v ∈ B(R), kPV ⊥ S(u) − PV ⊥ S(v)k ≤ γku − vk. (P4) (a) K is compact if X is defined by (I), bounded if X is defined by (II). (b) Let V be given by (P3) with R = R0 . Then ν = (PV )∗ ν × (PV ⊥ )∗ ν where (PV )∗ ν has a density ρ with respect to the Lebesgue measure on V , Ω := {ρ > 0} has piecewise smooth boundary and ρ|Ω is Lipschitz. We remark that (P1)–(P3) are selected to reflect the properties of general (nonlinear) parabolic PDEs. Definition 3.1 A probability measure µ on H is called an invariant measure for X if Θ0 = µ implies Θn = µ for all n > 0. Lemma 3.1 Assume (P1), (P2) and (P4)(a). Then (i) X has an invariant measure; (ii) there exists a compact set A ⊂ B(R0 ) on which all invariant measures of X are supported.
6
Proof. Let A0 = B(R0 ). For n > 0, let An = S(An−1 ) + K in the case of (I) and An = S(An−1 +K) in the case of (II). Then each An is compact, and by (P2), An ⊂ A0 for all n ≥ some N0 . Let 0 −1 A = ∪N (∩∞ k=0 AkN0 +i ) . i=0
Then A is compact, contained in B(R0 ), and satisfies S(A) + K = A. To construct an invariant measure for X , pick an arbitrary u0 ∈ A, and let Θ0P= δu0 , the Dirac measure at u0 . Then any accumulation point of the sequence { n1 i 0, there exists τ = τ (σ) < 1 such that the following holds for every f : H → R of class C 0,σ : for µ-a.e. u0 , there exists C = C(f, u0 ) such that Z Z < Cτ n for all n ≥ 1. f dΘn − f dµ Let X n denote the n-step Markov chain associated with X .
Theorem A (Structure of invariant measures). Assume (P1)–(P4). Then (1) X has at most a finite number of ergodic invariant measures. (2) If (X n , µ) is ergodic for all n ≥ 1, then (X , µ) is exponentially mixing for H¨older continuous observables. The reasons behind these results are that under (P1)–(P4), X resembles a Markov chain on RN whose transition probabilities have densities. One expects, therefore, the same type of decomposition into ergodic and mixing components. We now give a condition that guarantees the uniqueness of invariant measures and other convergence properties. This condition is expressed in terms of the existence of special sequences of controls; it is quite strong, but is easily verified for the PDEs under consideration.
7
(C) Given ε0 > 0 and R > 0, there is a finite sequence of controls ηˆ0 , · · · ηˆn such that for all u0 , u′0 ∈ B(R), if uk+1 = S(uk ) + ηˆk and u′k+1 = S(u′k ) + ηˆk for k < n, then kun − u′n k < ε0 . Theorem B (Sufficient condition for uniqueness and mixing). (P1)–(P4) and (C). Then
Assume
(1) X has a unique invariant measure µ, and (X , µ) is exponentially mixing; (2) ∃τ = τ (σ) < 1 such that ∀f ∈ C 0,σ and for every u0 ∈ H, there exists C s.t. Z Z f dΘn − f dµ < Cτ n for all n ≥ 1.
Recalling that the invariant measure µ is supported on a (relatively small) compact subset of H, we remark that the assertion in (2) above is considerably stronger than the usual notion of exponential mixing: it tells us about initial conditions far away from the support of µ. This property is reminiscent of the idea of Sinai-RuelleBowen measures for attractors in finite dimensional dynamical systems.
3.2
Proofs of abstract results (Theorems A and B)
We will prove Theorems A and B for the case where X is defined by (I); the proofs for (II) are very similar. Also, to avoid the obstruction of main ideas by technical details, we will assume (PV )∗ ν is the normalized Lebesgue measure on Ω := {u ∈ V, kuk ≤ r} for some r > 0; the general case is messier but conceptually not different. Let M = MR0 where R0 is given by (P2) and MR0 is as defined in (P1). The following notation is used heavily: Given u0 and η = (η0 , η1 , η2 , · · · ) ∈ K N , we define ui(η) inductively by letting u0 (η) = u0 and ui (η) = S(ui−1 (η)) + ηi−1 for i > 0. Notation such as ui (η0 , · · · , ηn−1 ) for a finite sequence (η0 , · · · , ηn−1 ) with i ≤ n has the obvious meaning, as does u′i(η) for given u′0 . Lemma 3.2 (Matching Lemma) Let δ = r(2M)−1 . There is a set Γ ⊂ K N with ν N (Γ) > 0 such that ∀u0 , u′0 ∈ B(R0 ) with ku0 −u′0 k < δ, there is a measure-preserving map Φ : Γ → K N with the property that ∀η ∈ Γ, kun (η) − u′n (Φ(η))k ≤ ku0 − u′0 kγ n
∀n ≥ 0 .
By virtue of (P4)(b), K = Ω × E where Ω ⊂ V is as above and E ⊂ V ⊥ . We write η0 = (η01 , η02 ) with η01 ∈ Ω, η02 ∈ E. Since all of our operations take place in V , it is convenient to introduce the notation Kε := {u ∈ Ω, kuk ≤ ε} × E, so that in particular Kr = K.
8
Proof. Suppose ku0 − u′0 k < δ. We define Φ(1) : K r2 = Kr−M δ → H by 1
2
(η0′ , η0′ ) = Φ(1) (η0 ) := (η01 + PV S(u0 ) − PV S(u′0 ), η02 ) . Observe that (i) kη0′ 1 k < (r − Mδ) + M · ku0 − u′0 k < r, so that Φ(1) (K 2r ) ⊂ K; (ii) Φ(1) preserves ν-measure; and (iii) for η0 ∈ K 2r , if u1 = u1 (η0 ) and u′1 = u′1 (Φ(1) (η0 )), then PV u1 = PV u′1
and
kPV ⊥ u1 − PV ⊥ u′1 k < γku0 − u′0 k.
We may, therefore, repeat the argument above with (u1 , u′1) in the place of (u0 , u′0 ), defining for each u1 = u1 (η0 ), η0 ∈ K 2r , a map from Kr−M δγ = Kr(1− 1 γ) to K. Put 2 together, this defines an injective map Φ(2) : K 2r × Kr(1− 1 γ) → K 2 which carries ν 2 2 measure to ν 2 -measure such that for each (η0 , η1 ) ∈ K r2 × Kr(1− 1 γ) , if u2 = u2 (η0 , η1 ) 2 and u′2 = u′2 (Φ(2) (η0 , η1 )), then PV u2 = PV u′2 and kPV ⊥ u2 − PV ⊥ u′2 k < γ 2 ku0 − u′0 k. Continued ad infinitum, this process defines a map Φ : Γ := K 2r × Kr(1− 1 γ) × Kr(1− 1 γ 2 ) × · · · → K N 2
2
with the desired properties. Clearly, ν(Γ) = Πi≥0 (1 − 21 γ i )D > 0 where D =dimV . Proof of Theorem A(1). Recall that ifPµ is an ergodic invariant measure for X , then by the Birkhoff Ergodic Theorem, n1 0n−1 δui (η) → µ for µ-a.e. u0 and ν N -a.e. η = (η0 , η1 , · · · ). This together with Lemma 3.2 implies that if µ, and µ′ are ergodic measures and there exist u0 ∈ supp(µ) and u′0 ∈ supp(µ′ ) with ||u0 − u′0 k < δ, then µ = µ′ . Since all invariant measures of X are supported on the compact set A (Lemma 3.1), it follows that there cannot be more than a finite number of them. Proof of the uniqueness of invariant measure part of Theorem B. From the last paragraph, we know that all the ergodic components of µ are pairwise ≥ δ apart in distance. Thus condition (C) with ε0 = δ and R = R0 guarantees that there is at most one ergodic component. We remark that the uniqueness of invariant measure results in Theorem 1 and Theorem 3(c) follow immediately from the preceding discussion once the abstract hypotheses (P1)–(P4) and (C) are checked for these equations. The next lemma is used only to prove the general result in Theorem A(2); it is not needed for the applications in Theorems 1–3. (Both the Navier-Stokes and Ginzberg-Landau equations satisfy much stronger conditions, making this argument unnecessary.) Let B(u, ε) denote the ball of radius ε centered at u, and let P n (·|u) denote the n-step transition probability given u. In the language introduced earlier, if Θ0 = δu , then P n (·|u) = Θn (·). 9
Lemma 3.3 Let µ be an invariant measure with the property that (X n , µ) is ergodic for all n ≥ 1. We fix B = B(˜ u, ε˜) where u˜ ∈ supp µ and ε˜ > 0. Then there exist + N0 ∈ Z and α0 > 0 such that P N0 (B|u) ≥ α0 for every u ∈ supp µ. Proof. Pick arbitrary u0 ∈ suppµ. Until nearly the end of the proof, the discussion ˆ n defined by pertains to this one point. Consider the “restricted distribution” Θ ˆ n (G) = ν n {(η0 , · · · , ηn−1 ) : ηi ∈ Kr−M δγ i ∀i < n and un (η0 , · · · , ηn−1 ) ∈ G} Θ ˆ n. where δ is as in Lemma 3.2, and let Wn denote the support of Θ Claim 1. d(u0, ∪n>0 Wn ) = 0. Pn−1 ˆ ˆ i converges weakly Proof. By compactness, a subsequence of n1 i=0 (Θi (A))−1 Θ to a probability measure µ ˜ on A (where A is as in Lemma 3.1). Since the restrictions on ηi become milder and milder as i → ∞, µ ˜ is an invariant measure for X . By ˆ construction, all the Θi are supported on supp µ, so we must have µ ˜ = µ, for we know from Theorem A(1) that all the other ergodic invariant measures have their supports bounded away from supp µ. Let N = N(u0 ) be such that d(u0 , WN ) < ε where ε < δ is a small positive number to be determined. Claim 2. For all k ≥ 0 and u ∈ WkN , ∃u′ ∈ W(k+1)N such that ku − u′ k < γ kN ε. Proof. The claim is true for k = 0 by choice of N. We prove it for k = 1: Let ∈ WN be such that ku0 −u′0 k < ε, and fix an arbitrary u ∈ WN . By definition, there exist ηi ∈ Kr−M δγ i such that u = uN (η0 , · · · , ηN −1 ). We wish to use the proximity ′ of u′0 to u0 and the Matching Lemma to produce (η0′ , · · · , ηN −1 ) with the property ′ ′ ′ ′ N that uN (η0 , · · · , ηN −1 ) ∈ W2N and kuN − uN k < εγ . To obtain the first property, it is necessary to have ηi′ ∈ Kr−M δγ i+N for all i < N. We proceed as follows: since ku0 − u′0 k < ε and η0 ∈ Kr−M δ , ∃η0′ ∈ Kr−M δ+M ε such that ku1 (η0 ) − u′1 (η0′ )k < εγ; similarly ∃η1′ ∈ Kr−M δγ+M εγ such that ku2 (η0 , η1 ) − u′1 (η0′ , η1′ )k < εγ 2 , and so on. (See the proof of Lemma 3.2.) Thus ηi′ ∈ Kr−M γ i (δ−ε) , and assuming ε is sufficiently small that δγ N < (δ − ε), we have ηi′ ∈ Kr−M δγ i+N . To prove the assertion for k = 2, we pick an arbitrary u ∈ W2N , which, by definition, is equal to vN from some v0 ∈ WN . Since we have shown that there exists v0′ ∈ W2N with kv0 − v0′ k < γ N ε, it suffices to ′ ′ repeat the argument above to obtain vN ∈ W3N with kvN − vN k < γ 2N ε. ˆ kN (B(˜ Claim 3. There exists k1 = k1 (u0) s.t. for k ≥ k1 , P kN (B|u) ≥ Θ u, 2ε˜ )) > 0 for all u ∈ H with ku − u0 k < δ. u′0
Proof. Let N (W, ε) denote the P ε-neighborhood of W ⊂ H. If follows from Claim 2 that if NkN := N (WkN , 2ε ki=0 γ iN ), then NkN ⊂ N(k+1)N for all k. Moreover, the ergodicity of (X N , µ) together with an observation similar to that in Claim 1 shows that the closure of ∪k NkN contains suppµ. Thus NkN ∩ B(˜ u, 4ε˜ ) 6= ∅ for large P∞ iN ˆ kN (B(˜ enough k. If 2ε i=1 γ < 4ε˜ , then Θ u, 2ε˜ )) > 0. Now for u with ku − u0 k < δ, 10
ˆ n starting from u0 can be coupled to a part of the the entire restricted distribution Θ (unrestricted) distribution starting from u. Thus for sufficiently large n, P n (B|u) ≥ ˆ n (B(˜ Θ u, 2ε˜ )). (1)
(n)
To finish, we cover supp µ with a finite number of δ-balls centered at u0 , · · · , u0 , ˆ where kˆ1 = maxi k1 (u(i) ) and N ˆ = Πi N(u(i) ). The lemma is and choose N0 = kˆ1 N 0 0 ˆ N0 (B(˜ ˆ N0 is the restricted distribution starting proved with α0 = mini Θ u, 2ε˜ )) where Θ (i) from u0 . From Lemma 3.2, we see that associated with each pair of points (u0 , u′0) with ku0 − u′0 k < δ, there is a cascade of matchings between un and u′n , leading to the definition of a measure-preserving map Φ : Γ := K r2 × Kr(1− 1 γ) × Kr(1− 1 γ 2 ) × · · · → K N 2
2
with the property that for η ∈ Γ, kui (η) − u′i (Φ(η))k ≤ γ i ku0 − u′0 k
for all i ≤ n.
The main goal in the next proof is, in a sense, to extend Φ to all of K N by attempting repeatedly to match the orbits that have not yet been matched. Proof of Theorem A(2). We consider for simplicity the case N0 = 1. Let u0 , u′0 ∈ supp µ, and let Θn and Θ′n denote the distributions of un and u′n respectively. We seek to define a measure-preserving map Φ : K N → K N and to estimate the difference between Θn and Θ′n by Z Z Z ′ |f (un (η)) − f (u′n (Φ(η)))| dν N (η) . In := f dΘn − f dΘn ≤
Let B be a ball of diameter δ centered at some point in suppµ. By Lemma 3.3, P (B|u0) ≥ α0 , and P (B|u′0 ) ≥ α0 . Matching u1 ∈ B to u′1 ∈ B, we define a measure˜ 1 → K for some Γ ˜ 1 ⊂ K with |Γ ˜ 1 | = α0 . This extends, by the preserving map Φ(1) : Γ ˜ 1 × Γ → K N . The map Matching Lemma, to a measure-preserving map Φ : Γ1 = Γ Φ|Γ1 represents the cascade of future couplings initiated by Φ(1) . Suppose now that Φ has been defined on ∪k≤n Γk where Γk is the set of η matched at step k. More precisely, Γ1 , Γ2 , · · · , Γn are disjoint subsets of K N , and each Γk is of ˜ k × Γ for some Γ ˜ k ⊂ K k ; the matching of uk and u′ in B that takes the form Γk = Γ k ˜ k → K k , while the cascade of future matchings place at step k defines a map Φ(k) : Γ ˜ k × Γ → K N . We now explain how to initiated by Φ(k) results in the definition of Φ : Γ (n) (n) ˜ n = K n \ ∪k≤n Γ where Γ = Γ ˜ k × Γ(n−k−1) is the first n-factors define Γn+1 . Let G k k ˜ n+1 defined by (η0 , · · · , ηn−1 ) ∈ G ˜ n; in Γk . Consider the restricted distribution Θ ′ ˜ the corresponding distribution Θn+1 is defined similarly. By Lemma 3.3, an α0 fraction of these two distributions can be matched, defining an immediate matching 11
˜ n+1 → K n+1 with Γ ˜ n+1 ⊂ G ˜ n × K and |Γ ˜ n+1 | = α0 |G ˜ n |. Future couplings Φ(n+1) : Γ (n+1) N ˜ n+1 × Γ. that result from Φ define Φ : Γn+1 → K with Γn+1 = Γ n ˜ We claim that ν (Gn ) decreases exponentially. This requires a little argument, for even though at each step a fraction of α0 of what is left is matched, our matchings (n) are “leaky”, meaning not every orbit defined by a sequence in Γk can be matched to ˜ n ), we write K N \∪k≤n Γk something reasonable at the (n+1)st step. To estimate ν n (G N ˜ n × K . The dynamics of (Gn , Hn ) → as the disjoint union Gn ∪ Hn where Gn = G (Gn+1 , Hn+1) are as follows: An α0 -fraction of Gn leaves Gn at the next step; of this part, a fraction of Πi≥0 (1 − 12 γ i )D (recall that D is the dimension of V ) goes into Γn+1 (see Lemma 3.2) while the rest goes into Hn+1 . At the same time, a fraction of Hn returns to Gn+1 . We claim that this fraction is bounded away from zero for all n. (n) To see this, consider one Γk at a time, and observe (from the definition of Γk ) that (n) (n+1) ˜ k |γ n−k . |(Γk × K) \ Γk | ∼ const |Γ Combinatorial Lemma Let a0 , b0 > 0, and suppose that an and bn satisfy recursively an+1 ≥ (1 − α0 )an + α1 bn
and
bn+1 ≤ (1 − α1 )bn + α0 an
for some 0 < α0 , α1 < 1. Then there exits c > 0 such that
an bn
> c for all n.
The proof of this purely combinatorial lemma is left as an exercise. We deduce ˜ n ) ≤ Cβ n for some C > 0 and from it that inf n |Gn |/|Hn | > 0, which implies ν n (G β < 1. This in turn implies that |Γn+1| ≤ Cβ n . Proceeding to the final count, we let f : supp µ → R be such that |f | < C1 and |f (u) − f (v)| < C1 ku − vkσ . Then In ≤
Z
˜n G
n
Z
|f (un (η0 , · · · , ηn−1 ))|dν + |f (u′n (η0 , · · · , ηn−1 ))|dν n (n) (n) n K −Φ (∪k≤n Γk ) X Z |f (u(η0 , · · · , ηn−1 )) − f (u′n (Φ(n) (η0 , · · · , ηn−1)))|dν n (6) + (n)
k≤n
Γk
≤ 2C1 · Cβ n +
X
Cβ k−1 · C1 (δγ n−k )σ
k≤n
≤ const n · [max(β, γ σ )]n ≤ const · τ n . Since these estimates are uniform for all pairs u0 , u′0, we obtain by integrating over u′0 that Z Z ≤ const · τ n . f dΘn − f dµ
12
Proof of Theorem B. We will prove, in the next paragraph, that assertion (2) in Theorem B holds for any invariant measure µ of X . From this (1) follows immediately: since (X , µ) is exponentially mixing, it is ergodic; and since µ is chosen arbitrarily, it must be the unique invariant measure. To prove the claim above, we pick arbitrary u0 ∈ H, u′0 ∈ A, and compare their distributions Θn and Θ′n as we did in the proof of Theorem A(2). First, by waiting a suitable period, we may assume that Θn is supported in B(R0 ) (where R0 is as in (P2)). By condition (C) with ε0 = δ where δ is as in Lemma 3.2, there is a set of controls of length N0 and having ν N0 -measure α0 for some α0 R> 0 that steer the R ˆ n| entire ball B(R0 ) into a set of diameter < δ. The estimate for | f dΘn − f dΘ now proceeds as in Theorem A(2), with the use of these special controls taking the place of Lemma 3.3 to guarantee that an α0 -fraction of what is left is matched every N0 steps. Averaging u′0 with respect to µ, we obtain the desired result.
3.3
Applications to PDEs: Proofs of Theorems 1 and 3
In this subsection, we prove the theorems related to PDEs stated in Sect. 2.1. Proof of Theorem 1. We will prove that the abstract hypotheses (P1)–(P4) and (C) hold for the incompressible Navier-Stokes equation in L2 for the type of noise specified. Let S(u0) = u(t = 1) where u is the solution of the Navier-Stokes equation with initial data u0 , and let uk = S(uk−1 ) + ηk . Most of the computations below are classically known (see for instance [2], [14]); we include them for completeness. We start by recalling a few properties of the Navier-Stokes equation in the 2-D torus. First, the following energy estimate holds for all t > 0: Z t 1 1 2 (7) ||u(t)||L2 + ν ||∇u||2L2 = ||u0||2L2 . 2 2 0 R Since u = 0, we have the Poincare inequality ||∇u||L2 ≥ ||u||L2 .
(8)
||S(u)||L2 ≤ e−ν ||u||L2 ;
(9)
From (7) and (8), it follows that
thus (P2) is satisfied by taking R0 (a) > 1−e1−ν a. On the other hand, for any two solutions u and v with initial conditions u0 and v0 , we have Z 1 2 2 ∂t ||u − v||L2 + ν||∇(u − v)||L2 ≤ | (u − v).∇v(u − v)| 2 (10) ≤ C||∇v||L2 ||u − v||L2 ||u − v||H 1 C ν ||u − v||2H 1 + ||∇v||2L2 ||u − v||2L2 . ≤ 2 ν 13
(H¨older and Sobolev inequalities are used to get the second line, and the CauchySchwartz inequality is used to get the third.) Then, applying a Gronwall lemma, we get Z 1
||S(u0) − S(v0 )||2L2 + ν
0
||(u − v)(s)||2H 1 ds ≤ CR ||u0 − v0 ||2L2 .
(11)
Here and below, CR denotes a generic constant depending only on R, an upper bound on the L2 norm of u0 , and on the viscosity ν. (P1)(b) follows from (11). To prove that (P3) holds, we use (11), (7) and a Chebychev inequality to deduce the existence of a time s, 0 < s < 1, such that ν||(u − v)(s)||2H 1 ≤ 4CR ||u0 − v0 ||2L2 , ν||u(s)||2H 1 < 2R2 and ν||v(s)||2H 1 < 2R2 . Combining these estimates with energy estimates in H 1 for t > s, namely, Z t 1 1 2 (12) ||∇u(t)||L2 + ν ||∆u||2L2 = ||∇u(s)||2L2 , 2 2 s Z t 1 1 2 (13) ||∇v(t)||L2 + ν ||∆v||2L2 = ||∇v(s)||2L2 , 2 2 s 1 (14) ∂t ||u − v||2H 1 + ν||u − v||2H 2 ≤ ||u − v||H 2 ||u − v||H 1 ||u||H 2 + ||v||H 2 2 1 ν ||u − v||2H 2 + ||u||2H 2 + ||v||2H 2 ||u − v||2H 1 , ≤ 4 ν integrating (14) between s and 1 and using again a Gronwall lemma, we deduce easily that (15) ||S(u0) − S(v0 )||H 1 ≤ CR ||u0 − v0 ||L2 . For any γ > 0 and R > 0, we may take N large enough that if VN :=span{e1 , e2 , ..., eN }, then CR ||u||L2 ≤ γ||u||H 1 ∀u ∈ VN⊥ . This together with (15) proves (P3). Finally, property (C) is satisfied by taking ηi = 0 for 1 ≤ i ≤ n0 where n0 is large enough that Re−νn0 ≤ ε0 (see (9)). The product structure of the noise ν 3 in property (P4)(b) holds because ξjk in (2) are independent; the assumption on PV ∗ ν holds because bj 6= 0 for 1 ≤ j ≤ N where N is as in (P3) and the law for ξjk has density ρj . Proof of Theorem 1’. We now prove (P1)–(P4) and (C) in H s . To prove (P1)(b), we use the energy estimates 1 ∂t ||u||2H s + ν||u||2H s+1 ≤ C||u||H s ||u||H s+1 ||u||H 1 2 C ν ||u||2H s+1 + ||u||2H 1 ||u||2H s , ≤ 2 ν 3
(16)
We hope our dual use of the symbol ν as viscosity and as noise does not lead to confusion.
14
1 ∂t ||u − v||2H s + ν||u − v||2H s+1 ≤ C||u − v||H s ||u − v||H s+1 (||u||H s+1 + ||v||H s+1 ) (17) 2 ν C ≤ ||u − v||2H s+1 + (||u||2H s+1 + ||v||2H s+1 )||u − v||2H s , 2 ν and Gronwall’s lemma between times 0 and 1. To prove (P3), we proceed as in the case of L2 , showing the existence of a time τ , 0 < τ < 1, such that ||(u−v)(τ )||H s+1 ≤ 4CR ||u0 −v0 ||H s and ||u(τ )||H s+1 , ||v(τ )||H s+1 ≤ 4CR where ||u0 ||H s , ||v0||H s < R. Then using (16) and (17) with s replaced by s + 1 and integrating between τ and 1, we deduce that ||S(u0) − S(v0 )||H s+1 ≤ CR ||u0 − v0 ||H s ,
(18)
from which we obtain (P3). To prove (P2), we make use of the regularizing effect of the Navier-Stokes equation in 2-D (19) ||S(u0)||H s ≤ Cs (||u||L2 ) where Cs is a function depending only on s (see [14]). Since BH s (a) ⊂ BL2 (a), we know from (P3) for L2 that if u0 ∈ BL2 (R), we have un ∈ BL2 (R0 ) ∀n ≥ some N0 . Taking Rs = Cs (R0 ) + a, we get that un ∈ BH s (Rs ) ∀n ≥ N0 . To prove (C), we argue as in L2 , taking ηi = 0, 1 ≤ i ≤ n0 , for large enough n0 and appealing to the fact that Cs (r) → 0 as r → 0. We remark that (P2) and (C) above can be proved directly without going through L2 . Next we move on to the real Ginzburg-Landau equation. Proof of Theorem 3. For simplicity, we take ν = 1. (a) We need to prove that there exist two disjoint stable sets A1 and A−1 , stable in the sense that ∀u ∈ A±1 , S(u) + η ∈ A±1 ∀η ∈ K. Let A1 = {u ∈ H, ||u − 1||L2 ≤ β}
(20)
where β is a constant to be determined. We recall for each φ ∈ R the energy estimate Z 1 2 2 (21) ∂t ||u − φ||L2 + ||∇(u − φ)||L2 + u(u − 1)(u + 1)(u − φ) dx = 0 . 2 T Substituting φ = 1 in (21), we get 1 ∂t ||u − 1||2L2 + ||∇(u − 1)||2L2 ≤ − 2
Z
u(u + 1)(u − 1)2 dx.
(22)
T
Now for any φ with 0 < φ < 1, we have u(u + 1)(u − 1)2 ≥ φ(φ + 1)(u − 1)2 u(u + 1)(u − 1)2 ≥ −1 15
if
u ≥ φ or u ≤ −1 − φ, ∀u.
(23)
Hence Z Z 2 u(u + 1)(u − 1) dx ≥ (1{u≥φ} + 1{u≤−1−φ} )φ(φ + 1)(u − 1)2 − meas{u ≤ φ}. T
T
Since the first term on the right side is Z 2 1{−1−φ 1 can be relaxed.
4 4.1
Dynamics with Negative Lyapunov Exponents Formulation of abstract results
We consider a semi-group St on H and a Markov chain X defined by (I) or (II) in the beginning of Sect. 3.1. In order for Lyapunov exponents to make sense, we need to impose differentiability assumptions. (P1’) (a) S(B(R)) is compact ∀R > 0; (b) S is C 1+Lip , meaning for every u ∈ H, there exists a bounded linear operator Lu : H → H with the property for all h ∈ H, 1 lim {S(u + εh) − S(u) − Lu (εh)} = 0 ε→0 ε
(34)
and ∀R > 0, ∃MR such that ∀u, v ∈ B(R), kLu − Lv k ≤ MR ku − vk. Since Lemma 3.1 clearly holds with (P1) replaced by (P1’), we let A be as in Section 3. 17
Proposition 4.1 Assume (P1’), (P2) and (P4)(a), and let µ be an invariant measure for X . Then there is a measurable function λ1 on H with −∞ ≤ λ1 < ∞ such that for µ-a.e. u0 and ν N -a.e. η = (η0 , η1 , η2 , · · · ), lim
n→∞
1 log kLun−1 ◦ · · · ◦ Lu1 ◦ Lu0 k = λ1 (u0 ) . n
Moreover, λ1 is constant µ-a.e. if (X , µ) is ergodic. This proposition follows from a direct application of the Subadditive Ergodic Theorem [6] together with the boundedness of kLu k on A (see also Lemma 4.1 below). We will refer to the function or, in the ergodic case, number λ1 as the top Lyapunov exponent of (X , µ). This section is concerned with the dynamics of X when λ1 < 0. We begin by stating a result, namely Theorem C, which gives a general description of the dynamics when λ1 < 0. This result, however, is not needed for our application to PDEs. The proof of Theorem 2 uses only Theorem D, which is independent of Theorem C. Let µ be an invariant measure of X . Theorem C concerns the conditional measures of µ given the past. That is to say, we view X as starting from time −∞, i.e. consider · · · , u−2 , u−1, u0 , u1 , u2, · · · defined by un+1 = Sun + ηn ∀n ∈ Z where · · · , η−2 , η−1 , η0 , η1 , η2 , · · · are ν-i.i.d. Then for ν Z -a.e. η = (· · · , η−1 , η0 , η1 , · · · ), the conditional probability of µ given η − := (· · · , η−2 , η−1 ) is well defined. We denote it by µη . Theorem C (Random sinks). Assume (P1’), (P2) and (P4)(a), and let µ be an ergodic invariant measure with λ1 < 0. Then there exists k0 ∈ Z+ such that for ν Z -a.e. η ∈ K Z , µη is supported on exactly k0 points of equal mass. This result is well known for stochastic flows in finite dimensions (see [11]). In the next theorem we impose a condition slightly stronger than (C) in Sect. 3.1 to obtain the type of uniqueness result needed for Theorem 2. (C’) There exists uˆ0 ∈ H such that for all ε0 > 0 and R > 0, there is a finite sequence of controls ηˆ0 , · · · ηˆn such that for all u0 ∈ B(R), if uk+1 = Suk + ηˆk and uˆk+1 = S uˆk + ηˆk for all k < n, then kun − uˆn k < ε0 . For u ∈ H, we define the accessibility set A(u) as follows: let A0 (u) = {u}, An (u) = S(An−1 (u)) + K for n > 0, and A(u) = ∪n≥0 An (u). Theorem D (Asymptotic uniqueness of solutions independent of initial condition). Assume (P1’), (P2), (P4)(a) and (C’). Suppose there is an ergodic invariant measure µ supported on A(ˆ u0 ) for which λ1 < 0. Then µ is the only invariant measure X has, and the following holds for ν N -a.e. η = (η0 , η1 , · · · ): ∀u0 , u′0 ∈ H,
kun (η) − u′n (η)k ≤ Ceλn 18
∀n > 0
where λ is any number > λ1 and C = C(u0 , u′0, λ). Roughly speaking, Theorem D allows us to conclude that all the orbits are eventually “the same” once we know that the linearized flows along some orbits are contractive. This passage from a local to a global phenomenon is made possible by condition (C’), which in the abstract is quite special but is satisfied by a number of standard parabolic PDEs.
4.2
Proofs of abstract results (Theorems C and D)
Let A be the compact set in Lemma 3.1, and let K denote the support of ν as before. We consider the dynamical system F : K N × A → K N × A defined by F (η, u) = (ση, S(u) + η0 ) where η = (η0 , η1 , η2 , · · · ) and σ is the shift operator, i.e. σ(η0 , η1 , η2 , · · · ) = (η1 , η2 , · · · ). The following is straightforward. Lemma 4.1 Let µ be an invariant measure of X in the sense of Definition 3.1. Then F preserves ν N × µ, and (F, ν N × µ) is ergodic if and only if (X , µ) is ergodic in the sense of Definition 3.2. Our next lemma relates the top Lyapunov exponent of a system, which describes the average infinitesimal behavior along its typical orbits, to the local behavior in neighborhoods of these orbits. A version applicable to our setting is contained in [13]. Let B(u, α) = {v ∈ H, kv − uk < α}. Proposition 4.2 [13] Let µ be an invariant measure, and assume that λ1 < 0 µ-a.e. Then given ε > 0, there exist measurable functions α, γ : K N × A → (0, ∞) and a measurable set Λ ⊂ K N × A with (ν N × µ)(Λ) = 1 such that for all (η, u0 ) ∈ Λ and v0 ∈ B(u0 , α(η, u0)), kvn (η) − un (η)k < γ(η, u0) e(λ1 +ε)n
∀n ≥ 0 .
We first prove Theorem D, from which Theorem 2 is derived. Proof of Theorem D. From (P2), it follows that we need only to consider initial conditions in B(R0 ). Fix ε > 0 and let α and Λ be as in Proposition 4.2 for the dynamical system (F, ν N × µ). We make the following choices: 99 . Covering (1) Let α0 > 0 be a number small enough that (ν N × µ){α > 2α0 } > 100 1 the compact set A(ˆ u0 ) with a finite number of 2 α0 -balls, we see that there exists u˜0 ∈ A(ˆ u0 ) such that
Γ1 := {η ∈ K N : B(˜ u0 , α0 ) ⊂ B(u, α(η, u)) for some u with (η, u) ∈ Λ} has positive ν N -measure. 19
(2) Since u˜0 ∈ A(ˆ u0 ), there is a sequence of controls (˜ η0 , · · · , η˜k−1) that puts uˆ0 in B(˜ u0 , 21 α0 ). Choose δ > 0 and Γ2 ⊂ K k with ν k (Γ2 ) > 0 such that if u0 ∈ B(ˆ u0 , δ) and (η0 , · · · , ηk−1) ∈ Γ2 , then uk (η0 , · · · , ηk−1 ) ∈ B(˜ u0 , α0 ). (3) Condition (C’) guarantees that there exists a sequence of controls (ˆ η0 , · · · , ηˆj−1 ) 1 j that puts the entire ball B(R0 ) inside B(ˆ u0 , 2 δ). Choose Γ3 ⊂ K with ν j (Γ3 ) > 0 such that every sequence (η0 , · · · , ηj−1) ∈ Γ3 puts B(R0 ) inside B(ˆ u0 , δ). Let Γ ⊂ K N be the set defined by {(η0 , · · · , ηj−1) ∈ Γ3 ; (ηj , · · · , ηj+k−1) ∈ Γ2 ; (ηj+k , ηj+k+1, · · · ) ∈ Γ1 } . Clearly, ν N (Γ) > 0. The following holds for ν N -a.e. η : Fix η, and let Bn denote the nth image of B(R0 ) for this sequence of kicks. By the ergodicity of (σ, ν N ), there exists N such that σ N η ∈ Γ. Choosing N ≥ N0 (R0 ), we have, by (P2), that BN ⊂ B(R0 ). The choice in (3) then guarantees that BN +j ⊂ B(ˆ u0 , δ), and the choice in (2) guarantees that BN +j+k ⊂ B(˜ u0 , α0 ). By (1), BN +j+k ⊂ B(u, α(u, σ N +j+k η)) N +j+k for some u with (σ η, u) ∈ Λ. Proposition 4.2 then says that when subjected to the sequence of kicks defined by σ N +j+k η, all orbits with initial conditions in BN +j+k converge exponentially to each other as n → ∞. Hence this property holds for all orbits starting from B(R0 ) when subjected to η. Theorem D is proved. Proceeding to Theorem C, the measures µη defined in Sect. 4.1 are called the sample or empirical measures of µ. They have the interpretation of describing what one sees at time 0 given that the system has experienced the sequence of kicks η − = (· · · , η−2 , η−1 ). The characterization of µη in the next lemma is useful. We introduce the following notation: Let Sη0 : H → H be the map defined by Sη0 (u) = Su + η0 ; for a measure µ on H, Sη0 ∗ µ is the measure defined by (Sη0 ∗ µ)(E) = µ(Sη−1 E). 0 Lemma 4.2 Let µ be an invariant measure for X . Then for ν Z -a.e. (· · · , η−2 , η−1 , η0 , ...), (Sη−1 Sη−2 · · · Sη−n )∗ µ converges weakly to µη . Proof: Fix a continuous function ϕ : A → R, and define ϕ(n) : K Z → R by Z (n) ϕ (η) = ϕ d((Sη−1 Sη−2 · · · Sη−n )∗ µ) Z = ϕ(Sη−1 Sη−2 · · · Sη−n (u))dµ(u).
η =
(35)
−n −n Then ϕ(n) is B−1 -measurableR where B−1 is the σ-algebra on K Z generated by coor−n+1 dinates η−1 , · · · , η−n . Since Sη−n ∗ µ dν(η−n ) = µ, we have E(ϕ(n) |B−1 ) = ϕ(n−1) . (n) The martingale convergence theorem then tells us that ϕ convergence ν Z -a.e. to −∞ a function measurable on B−1 . It suffices to carry out the argument above for a countable dense set of continuous functions ϕ.
20
Lemma 4.3 Given δ > 0, ∃N = N(δ) ∈ Z+ such that for ν Z -a.e. η, there is a set Eη consisting of ≤ N points such that µη (Eη ) > (1 − δ). Proof: Let α and γ be the functions in Proposition 4.2 for the dynamical system (F, ν N × µ). Given δ > 0, we let α0 , γ0 > 0 be constants with the property that if G = {(η, u) : α(η, u) ≥ α0 , γ(η, u) ≤ γ0 } and Γ = {η ∈ K N : µ{u : (η, u) ∈ G} > 1 − δ} , then ν N (Γ) > 1 − δ. Consider η ∈ K Z such that (i) µη = lim(Sη−1 Sη−2 · · · Sη−n )∗ µ and (ii) (η−n , η−n+1, · · · ) ∈ Γ for infinitely many n > 0. By Lemma 4.2 and the ergodicity of (σ, ν Z ), we deduce that the set of η satisfying (i) and (ii) has full measure. We will show that the property in the statement of the lemma holds for these η. Fix a cover {B1 , · · · , BN } of A by α20 -balls, and let η be as above. We consider n arbitrarily large with (η−n , η−n+1 , · · · ) ∈ Γ. For each i, 1 ≤ i ≤ N, such that Bi ∩ {u ∈ H : ((η−n , η−n+1 , · · · ), u) ∈ G} = 6 ∅, pick an arbitrary point u(i) in this set. Our choices of G and Γ ensure that µ(∪i B(u(i) , α0 )) > 1 − δ, and that the diameter of (Sη−1 Sη−2 · · · Sη−n )B(u(i) , α0 ) is ≤ γ0 α0 e(λ+ε)n . We have thus shown that a set of µη -measure > 1 − δ is contained in ≤ N balls each with diameter ≤ γ0 α0 e(λ+ε)n . The result follows by letting n → ∞. To prove Theorem C, we need to work with a version of (F, ν N ×µ) that has a past. Let F˜ : K Z × A → K Z × A be such that F˜ : (η, u) 7→ (ση, Sη0 u), and let ν Z ∗ µ be the measure which projects onto ν Z in the first factor and has conditional probabilities µη on η-fibers. That ν Z ∗ µ is F˜ -invariant follows immediately from Lemma 4.2. It is also easy to see that (F˜ , ν Z ∗ µ) is ergodic if and only if (F, ν N × µ) is. Proof of Theorem C. It follows from Lemma 4.3 that for ν Z -a.e. η, µη is atomic, with possibly a countable number of atoms. We now argue that there exists k0 ∈ Z+ such that for a.e. η, µη has exactly k0 atoms of equal mass. Let h(η) = supu∈H µη {u} . To see that h is a measurable function on K Z , let P (n) , n = 1, 2, · · · , be an increasing sequence of finite measurable partitions of A such that diamP (n) → 0 as n → ∞. Then for each P ∈ P (n) , η 7→ µη (P ) is a measurable function, as are hn := maxP ∈P (n) µη (P ) and h := limn hn . Observe that h(ση) ≥ h(η), with > being possible in principle since Sη0 is not necessarily one-to-one. However, the measurability of h together with the ergodicity of (σ, ν Z ) implies that h is constant a.e. Let us call this value h0 . From the last lemma we know that h0 > 0. 21
To finish, we let X = {(η, u) ∈ K Z × A : µη {u} = h0 }. Then X is a measurable set, (ν Z ∗ µ)(X) > 0 and F˜ −1 X ⊃ X. This together with the ergodicity of (F˜ , ν Z ∗ µ) implies that (ν Z ∗ µ)(X) = 1, which is what we want.
4.3
Application to PDEs: Proof of Theorem 2
Let St be the semi-group generated by the (unforced) Navier-Stokes system, and let S = S1 . Lemma 4.4 S is C 1+Lip in H 2 (R2 ). Proof. It is easy to see that Lu is defined by Lu w = ψ(1) where ψ is the solution of the linear problem ∂t ψ + U.∇ψ + ψ.∇U − ν∆ψ = −∇p, (36) ψ(t = 0) = w , divψ = 0 where U denotes the solution of the Navier-Stokes system with initial data u. That Lu is linear, continuous and goes from H 2 to H 2 is obvious. To prove that (34) holds, let U and V be the solutions of the Navier-Stokes system with initial data u and u + ǫw respectively. Then y = V − U − ǫψ satisfies ∂t y + (U + ǫψ).∇y + y.∇V + ǫ2 ψ.∇ψ − ∆y = −∇p (37) y(t = 0) = 0 , div(y) = 0. By a simple computation, we get that ||y(t = 1)||H 2 ≤ C(1 + ||w||2H 2 )ǫ2 , where here and below C denotes a constant depending only on the H 2 norm of u. To prove that Lu is Lipschitz, i.e., ||(Lu − Lv )w||H 2 ≤ C||u − v||H 2 ||w||H 2 ,
(38)
we define Lv w = φ(1) where φ solves an equation analogous to (36) with V in the place of U, V being the solution with initial condition v. The desired estimate ||(ψ − φ)(t = 1)||H 2 is obtained by subtracting this equation from (36). Remark. We observe here that the top Lyapunov exponent is negative if the noise is sufficiently small. We will show, in fact, that given any positive viscosity ν, if a (see Sect. 2.1 for definition) is small enough, then S : H 2 → H 2 is a contraction on ν the ball of radius 2C . Rewriting equations (16) and (17) with s = 2, we have ∂t ||u||2H 2 + ν||u||2H 3 ≤
22
C2 ||u||4H s , ν
(39)
∂t ||u − v||2H 2 + ν||u − v||2H 3 ≤
C2 (||u||2H 3 + ||v||2H 3 )||u − v||2H 2 . ν
(40)
ν ν For u0 with ||u0||H 2 ≤ 2C and a noise with a ≤ 2C (1 − e−ν/4 ), it follows from (39) and a Gronwall lemma that ν 2 ||S(u0)||2H 2 ≤ e−ν/2 , 2C
from which we obtain ||u1 ||H 2 ≤ ν
ν . 2C
Z
0
Moreover, from (39), we have that
1
||u||2H 3 ≤
ν3 , 16C 2
so that if v is another solution of the Navier-Stokes system with ||v0 ||H 2 ≤ (40) gives ν ||u − v||2H 2 ≤ ||u0 − v0 ||2H 2 e−ν+ 8 .
ν , 2C
then
Proof of Theorem 3. It suffices to check the hypotheses of Theorem D: (P1’) is proved in Lemma 4.4, and we explained in the proof of Theorem 1’ why (C’) holds with uˆ0 = 0.
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